Huong T L Vu

Frédéric J P Richard

Statistical tests of heterogeneity for anisotropic multifractional Brownian fields

Keywords: isotropy, anisotropy, homogeneity, heterogeneity, increment, quadratic variations, statistical test, anisotropic fractional Brownian field, multifractional Brownian field

In this paper, we deal with some anisotropic extensions of the multifractional Brownian fields that account for spatial phenomena whose properties of regularity and directionality may both vary in space. Our aim is to set statistical tests to decide whether an observed field of this kind is heterogeneous or not. The statistical methodology relies upon a field analysis by quadratic variations, which are averages of square field increments. Specific to our approach, these variations are computed locally in several directions. We establish an asymptotic result showing a linear Gaussian relationship between these variations and parameters related to regularity and directional properties of the model. Using this result, we then design a test procedure based on Fisher statistics of linear Gaussian models. Eventually we evaluate this procedure on simulated data.

Introduction

The history of the Brownian fields dates back to the introduction of the fractional Brownian motion by Kolmogorov and Mandelbrot [START_REF] Kolmogorov | Wienersche spiralen und einige andere interessante kurven im hibertschen raum[END_REF][START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF]. The spatial extension of this process has paved the way for a multitude of random field models to describe spatial phenomena of an irregular nature. These fields are roughly divided into two large families: anisotropic and heterogeneous fields. Anisotropic fields include anisotropic fractional Brownian fields [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF], their extension in the framework of intrinsic fields [START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF], or operatorscaling fields [START_REF] Benson | Aquifer operator-scaling and the effect on solute mixing and dispersion[END_REF][START_REF] Biermé | Operator scaling stable random fields[END_REF][START_REF] Clausel | Explicit construction of operator scaling Gaussian random fields[END_REF]. They enable to model homogeneous spatial phenom-H. Vu and F. Richard are at Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France. E-mail: [thi-lan-huong.vu,frederic.richard]@univ-amu.fr ena whose global properties vary according to directions. Heterogeneous fields mainly refer to the multifractional Brownian fields [START_REF] Benassi | Elliptic Gaussian random processes[END_REF][START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF] and its various extensions [START_REF] Ayache | Generalized multifractional Brownian motion: definition and preliminary results[END_REF][START_REF] Benassi | Identification of the Hurst index of a step Fractional Brownian motion[END_REF], but also to heterogeneous operator-scaling fields [START_REF] Biermé | Multi-operator scaling random fields[END_REF]. They permit to model heterogeneous spatial phenomena whose regularity may vary in a local way. Most of these fields are essentially isotropic. In this paper, we deal with more generic heterogeneous fields accounting for spatial phenomena whose regularity and anisotropy may both vary. The emergence of Brownian fields has raised many statistical issues. The main questions revolve around the estimation of parameters (possibly functional) of these fields (see, for instance, [START_REF] Istas | Quadratic variations and estimation of the local Hölder index of a Gaussian process[END_REF][START_REF] Biermé | Estimation of anisotropic Gaussian fields through Radon transform[END_REF][START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF][START_REF] Benassi | Identifying the multifractional function of a Gaussian process[END_REF][START_REF] Coeurjolly | Identification of multifractional Brownian motion[END_REF][START_REF] Ayache | On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion[END_REF]). Other questions concern tests of properties of these fields (anisotropy, heterogeneity, etc) [START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF][START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF]. A widespread approach to analyzing Brownian fields is based on quadratic variations, which are averages of squared field increments. This approach has been developed both in oriented forms to analyze anisotropic fields [START_REF] Biermé | Estimation of anisotropic Gaussian fields through Radon transform[END_REF][START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF] or local forms to analyze heterogeneous fields [START_REF] Benassi | Identifying the multifractional function of a Gaussian process[END_REF][START_REF] Coeurjolly | Identification of multifractional Brownian motion[END_REF]. The specification of this approach is usually associated to a convergence study that aims at identifying asymptotic laws of model estimates. The knowledge of this law has served as a theoretical basis for the construction of statistical estimates of parameters and hypothesis tests. In this paper, we develop a quadratic variation approach for the statistical analysis of anisotropic heterogeneous fields. We construct local estimates of field properties, and identify their asymptotic law. We eventually design some tests to check if a field is spatially homogeneous or not.

The heterogeneity of fields we consider may have different aspects. As illustrated in Figure 1 (a), it may be due to spatial variations of field regularity. As shown in Figures 1 (b) and (c), it can also result from local variations of directional field properties. These variations may further have different origins: for instance, variations of the degree of anisotropy as in Figure 1 (b), variations of pattern orientations as in Figure 1 (c). Eventually, the heterogeneity can reflect a combination of several types of variations. The purpose of our statistical tests is to help decide whether properties of the field are the same at several space locations in terms of regularity, anisotropy or both.

2 Brownian field models

Previous models

Derived from the fractional Brownian motion [START_REF] Kolmogorov | Wienersche spiralen und einige andere interessante kurven im hibertschen raum[END_REF][START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF], the fractional Brownian field is both isotropic and homogeneous. Its homogeneity is due to the stationarity of its increments, and provides it with a constant Hölder regularity all over the space.

In [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF], Bonami and Estrade introduced some anisotropic extensions of the fractional Brownian field. Among them, the so-called anisotropic fractional Brownian field is a field with stationary increments whose spectral density is of the form

∀ω ∈ R d , h τ,β (ω) = τ(ω)|ω| -2β(ω)-d , ( 1
)
where functions τ and β are called the topothesy and Hurst functions, respectively. By assumption, these functions are both homogeneous, i.e.

∀ w ∈ R d \{0}, τ x (w) = τ x w |w| and β x (w) = β x w |w| .

In other words, their values only depend on the direction w |w| of the frequency vector w. When they are both constant, the spectral density of Equation ( 1) is radial. In such a case, the associated field is isotropic and corresponds to a fractional Brownian field. At contrary, when one of these functions is not constant, the spectral density is non-radial and the field is anisotropic. Although more generic than the fractional Brownian fields, these anisotropic fields are still homogeneous. On the one hand, their increments are stationary. On the other hand, they have the same regularity and directional properties all over the space.

Following [START_REF] Benassi | Elliptic Gaussian random processes[END_REF], the multifractional Brownian field can be defined, for any position x, by a stochastic integral of the form

Z(x) = C x R d e i x,ω -1 |ω| H x +d/2 d W (ω) (2)
where d W stands for a complex Brownian measure, and C x and H x are two positive constants that may vary according to the position x. The Hölder regularity of this field is not constant anymore. Its order varies depending on H x . These variations were originally assumed to be Hölder so as to ensure that the field realization are continuous [START_REF] Benassi | Elliptic Gaussian random processes[END_REF]. This assumption was subsequently weakened to allow sharper variations of the regularity [START_REF] Ayache | Generalized multifractional Brownian motion: definition and preliminary results[END_REF][START_REF] Benassi | Identification of the Hurst index of a step Fractional Brownian motion[END_REF]. At a position x, a multifractional Brownian field admits a tangent field Zx which is a fractional Brownian field with a Hurst index H x . This tangent field is obtained as a limit [START_REF] Benassi | Elliptic Gaussian random processes[END_REF] ( Zx (u))

u∈R d = lim t→0 + Z(x + tu) -Z(x) t H x u∈R d .
According to this property, a multifractional Brownian field has the same local properties as a fractional Brownian field. Consequently, it is locally isotropic as its tangent field. Some anisotropic extensions of the multifractional Brownian field were considered in [START_REF] Polisano | Texture modeling by Gaussian fields with prescribed local orientation[END_REF][START_REF] Polisano | Modélisation de textures anisotropes par la transformée en ondelettes monogènes, et super[END_REF]. At any fixed position x of R d , they were defined by a stochastic integral

Z(x) = R d (e i x,ω -1) h τ x ,H x (ω) d W (ω) (3)
in which h τ x ,H x is a spectral density of the form (1) specified by a constant function H x and an homogeneous function β x that may both change from a position x to another. Polisano showed that such a field is tangent to an anisotropic fractional Brownian field with a spectral density h τ x ,H x at a position x [START_REF] Polisano | Modélisation de textures anisotropes par la transformée en ondelettes monogènes, et super[END_REF]. In other words, locally, it has the same directional properties as an anisotropic fractional Brownian field.

An extended model

In this paper, we propose a more generic anisotropic multifractional Brownian field Z (AMFBF) defined, for some A ≥ 0 and all positions x ∈ R d , by a stochastic integral

Z(x) = |w|>A (e i x,ω -1) f x (ω) d W (ω), (4)
where f x = h τ x ,β x is a spectral density of the form (1) specified by two homogeneous functions τ x and β x . In contrast with the work of [START_REF] Polisano | Texture modeling by Gaussian fields with prescribed local orientation[END_REF][START_REF] Polisano | Modélisation de textures anisotropes par la transformée en ondelettes monogènes, et super[END_REF], the Hurst functions β x are no longer constant. Moreover, the integral is partially defined for |w| > A where A is arbitrary. Therefore, the proposed field may not have the same lowest frequencies as a fractional Brownian field. For any x ∈ R d , we will assume that the functions τ x and β x are both homogeneous and even, i.e.

∀ w ∈ R d , τ x (w) = τ x (-w) = τ x w |w| and β x (w) = β x (-w) = β x w |w| .
This implies that both functions are determined by their values on the unit sphere

S = {u ∈ R d , |u| = 1} of R d .
H := inf x∈R d ess inf s∈S β x (s) > 0.
For any x ∈ R d , we will denote

H x = ess inf s∈S β x (s), (5)
and assume that the set

E x = {s ∈ S, β x (s) = H x and τ x (s) > 0} (6)
is of positive measure on S. Under this assumption, the parameter H x , called the Hurst index, gives the order of the local Hölder regularity of the field at x [START_REF] Benassi | Elliptic Gaussian random processes[END_REF]. By abusing the notation, H x will also refer to a function defined on R d and taking the constant value H x . We will also use a function δ x defined, for all w ∈ R d , by

δ x (w) = τ x (w) if β x (w) = H x , 0 otherwise. (7)
Besides, we will assume that the Hurst function x → β x is locally Hölder of an order η > β for the L ∞ -norm on S: i.e. for any x ∈ R d , there exist a positive constant K x and a local neighborhood

V x of x such that ∀ y, z ∈ V x , sup s∈S |β y (s) -β z (s)| ≤ K x |y -z| η . ( 8
)
This assumption (8) extends the one in [START_REF] Polisano | Modélisation de textures anisotropes par la transformée en ondelettes monogènes, et super[END_REF] to non-constant Hurst functions. We will also assume that the topothesy function x → τ x is locally Hölder of an order α > β for the L 2 -norm on S: i.e. for any x ∈ R d , there exist a positive constant Kx and a local neighborhood Ṽx of x such that

∀ y, z ∈ Ṽx , S τ 1/2 y (s) -τ 1/2 z (s) 2 ds ≤ Kx |y -z| α . ( 9
)
This assumption is also made in [START_REF] Polisano | Modélisation de textures anisotropes par la transformée en ondelettes monogènes, et super[END_REF].

The Hölder assumptions made on the Hurst and topothesy functions constrain their order of regularity to be larger than a bound β which is above the largest local Hölder regularity of the field.

Under the assumption above, we found the expression of tangent fields of our anisotropic multifractional Brownian field, extending the result in [START_REF] Polisano | Modélisation de textures anisotropes par la transformée en ondelettes monogènes, et super[END_REF].

Theorem 1 Let Z be a generic anisotropic multifractional Brownian field satisfying assumptions of this section. For any fixed x ∈ R d , we denote by Zx the anisotropic fractional Brownian defined, for all y ∈ R d , by

Zx (y) = R d (e i y,ω -1) h δ x ,H x (ω) dW (ω), ( 10
)
with a spectral density h δ x ,H x of the form (1) specified by the topothesy function δ x of Equation [START_REF] Benassi | Identification of filtered white noises[END_REF] and the Hurst function H x taking the constant value of Equation [START_REF] Benassi | Identifying the multifractional function of a Gaussian process[END_REF]. Then,

( Zx (u)) u∈R d = lim t→0 + Z(x + tu) -Z(x) t H x u∈R d
, the convergence being in distribution on the space of continuous functions endowed with the topology of the uniform convergence on compact sets [START_REF] Falconer | Tangent fields and the local structure of random fields[END_REF].

The field Zx is an homogeneous field which is called the tangent field of Z at x. It characterizes local properties of the field Z at x. Due to this property, Z is locally asymptotically self-similar (LASS) of order H x at X [START_REF] Benassi | Elliptic Gaussian random processes[END_REF].

Being zero-mean, an AMFBF can not model phenomena with large trends. To overcome this shortcoming, we will consider an additive random field model Z = Z L + Z H composed of an AMFBF Z H satisfying conditions of this section and an intrinsic random field Z L of order M ∈ N ∪ {-1} [START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF][START_REF] Matheron | The intrinsic random functions and their applications[END_REF][START_REF] Chilès | Geostatistics: modeling spatial uncertainty[END_REF]. The field Z L may include large polynomial trends of an order M + 1. We will further assume that it has a spectral density g which fulfills the condition

|ω| > B ⇒ g(ω) ≤ C|ω| -2 H-d (11)
for some B ≥ 0, C > 0, and H > β. This condition ensures that the intrinsic field is more regular than the AMFBF Z H at any point x of R d , and that highfrequencies of Z H are predominant in Z.

Analysis method

In this section, we propose a method to analyze AMFBF defined in Section 2.2. This method is based on localized quadratic variations that extend the ones used in [START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF]. We first define these quadratic variations. Then, we establish their asymptotic normality. Throughout this section, we will assume that a field Z is observed at points x = k/N for k varying on a grid T N = [[1, N ]] d and N fixed in N * . We will denote Z N [k] the observed field Z(x) at position x = k/N .

Quadratic variations

Quadratic variations are common tools for the analysis of processes or fields derived from the fractional Brownian motion. They were initially used to identify the Hurst index of fractional Brownian motions and some related processes [START_REF] Benassi | Identification of filtered white noises[END_REF][START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF][START_REF] Coeurjolly | Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths[END_REF][START_REF] Istas | Quadratic variations and estimation of the local Hölder index of a Gaussian process[END_REF]. They were also used to estimate the Hurst function of multifractional Brownian motions or fields [START_REF] Benassi | Identifying the multifractional function of a Gaussian process[END_REF][START_REF] Coeurjolly | Identification of multifractional Brownian motion[END_REF][START_REF] Istas | Quadratic variations and estimation of the local Hölder index of a Gaussian process[END_REF] and their generalizations [START_REF] Ayache | On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion[END_REF][START_REF] Ayache | A central limit theorem for the quadratic variations of the step fractional Brownian motion[END_REF][START_REF] Benassi | Identification of the Hurst index of a step Fractional Brownian motion[END_REF]. Besides, they were used to estimate regularity and directional features of anisotropic fractional Brownian fields and related fields [START_REF] Chan | Increment-based estimators of fractal dimension for twodimensional surface data[END_REF][START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF][START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF][START_REF] Richard | Some anisotropy indices for the characterization of Brownian textures and their application to breast images[END_REF][START_REF] Zhu | Parameter estimation for fractional Brownian surfaces[END_REF].

Quadratic variations of a field Z are averages of square increments of Z. When observed on a grid, an increment

V N [m] of Z is a linear combination of the form V N [m] = k∈Z d v[k]Z N [m -k], ( 12
)
where v is a finite support kernel v such that

k∈Z d v[k]P ( m-k N )
vanishes for any polynomial P of a fixed order K [START_REF] Chilès | Geostatistics: modeling spatial uncertainty[END_REF][START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF]. A common kernel is defined by

v[k] = (-1) |k| L 1 k 1 • • • L d k d , ( 13
)
for k ∈ [[0, L 1 ]] × • • • × [[0, L d ]] and v[k] = 0 otherwise. Such a kernel is of order K = |L 1 | + • • • + |L d | -1.
In what follows, we will denote

[[0, L]] = [[0, L 1 ]] × • • • × [[0, L d ]] the support of the kernel v.
For the local analysis of multifractional Brownian motions or fields, quadratic variations were usually computed in a neighborhood of a point x of R d (see [START_REF] Coeurjolly | Identification of multifractional Brownian motion[END_REF] for instance). With our notations, this operation amounts to sum up some increments of the form

V N [m] = k∈Z d v[k]Z N [m -k + p N x ], (14)
specifically centered at a grid point p N x getting closer and closer to x as the resolution N tends to +∞.

For the analysis of anisotropic fractional Brownian fields, quadratic variations were often associated to a directional transform of the field. In [START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF], they were computed on increments of the process resulting from a Radon projection of the field. In [START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF][START_REF] Richard | Some anisotropy indices for the characterization of Brownian textures and their application to breast images[END_REF], they involved oriented increments obtained by applying to the field some combinations T of rescaling and rotation of Z d into itself. These increments were of the form

V N [m] = k∈Z d v[k]Z N [m -T k]. ( 15
)
In dimension 2, the transform T were defined, for u ∈ Z 2 , as

T u = u 1 -u 2 u 2 u 1 = |u| cos(arg(u)) -sin(arg(u)) sin(arg(u)) cos(arg(u)) ,
which corresponds to a rescaling of factor |u| and a rotation of angle arg(u).

More generally, in dimension d ≥ 2, the transform can be expressed as T = ρR where ρ > 0 is a rescaling factor and R is a rotation matrix in R d satisfying R T R = I. Such a matrix R can be decomposed as a product of d -1 matrices (R (j) , j ∈ [[1, d -1]]) of R d accounting for rotations on successive planes (e 1 , e j+1 ). To R, we can associate a multi-dimensional angle ϕ = (ϕ (1) , • • • , ϕ (d-1) ) where ϕ (j) corresponds to the angle of the rotation R (j) in the plane (e 1 , e j+1 ).

In this paper, we propose to use increments that are both localized and oriented. These increments are of the form

V N [m] = k∈Z d v[k]Z N [m -T k + p N x ]. ( 16
)
We assume that the sequence (p N x ) N of grid points is chosen in such a way that

p N x N -x ∞ = O N →+∞ (N -1 ), (17)
taking, for instance, p N x = N x for k ∈ [ [1, d]]. We select a finite set of transforms T indexed in F and a finite set of positions x indexed in G. For (a, x) ∈ F × G, we will denote by V N a,x the increment field obtained with an ath transform and at an xth point. We will also denote T a , ρa , ϕ a the ath transform, the square of its rescaling factor and its rotation angle, respectively.

Next, for a fixed N , we define quadratic variations on a local neighborhood of each position x of G as

W N a,x = 1 n m∈V N (V N a,x [m]) 2 , ( 18
)
where

V N = [[-N , N ]] d is a set of n indices m ∈ Z d such that, for all (a, x) ∈ F × G, and k ∈ [[1, L]], m -T a k + p N
x is on the observation grid T N , and

m -T a k + p N x N -x ≤ N . ( 19
)
The parameter N determines the precision at which x is approached by grid points m-T a k+p N x N of V N . It has to satisfy three conditions:

(i) lim

N →+∞ N = 0, (ii) lim N →+∞ N N = +∞, (iii) lim N →+∞ log(N ) α N = 0, ∀ 0 < α < 1. (20)
Condition (i) makes the precision increase with N while condition (ii) enables to increase the cardinality of V N to +∞. Condition (iii) is required to ensure the asymptotic normality of quadratic variations (see Theorem 2 and its proof for details). To fulfill the three conditions, the parameter N can be of the form

N = KN -b log(N ) c , for any K > 0, 0 < b < 1 and c ∈ R.
For (a, x) ∈ F × G, we also define log-variations Y N a,x and log-scales ρ a as

Y N a,x = log(W N a,x
) and ρ a = log( ρa ), respectively.

Asymptotic normality

Next, we state a theorem about asymptotic properties of the log-variation random vector Y N = (Y N a,x ) (a,x)∈F ×G defined above. Theorem 2 Let Z = Z L + Z H be the sum of two independent fields: an anisotropic multifractional Brownian field Z H and an intrinsic random field Z L of order M satisfying assumptions of Section 2.2. Consider a log-variation vector Y N constructed using increments of an order K ≥ max(M, d 4 , 1). Let ζ N be a vector with terms

∀a ∈ F , x ∈ G, ζ N a,x = ρ a H x + B N x (ϕ a ), ( 22
)
where B N x is defined, for any angle ϕ, by

B N x (ϕ) = log C x (ϕ) N 2H ,
and

C x (ϕ) = 1 (2π) d +∞ 0 E x | v(ρR ϕ s)| 2 δ x (s)ρ -2H x -1 dsdρ, (23)
R ϕ standing for the matrix of rotation of angle ϕ.

Then, √ n (Y N -ζ N ) f .d.d. -→ N →+∞ N (0, Σ) (24)
for some covariance matrix Σ whose terms are given, for all (a, x), (b, y) ∈ F × G, by

Σ a,x;b,y = 2(2π) d [0,2π] d | ha,x;b,y (ω)| 2 dω ρ2H x a C x (ϕ a ) ρ2H x b C y (ϕ b ) , ( 25
)
where ha,x;b,y (ω) is defined, for any ω

∈ [0, 2π] d , by v(T a ω) v(T b ω) q∈Z d δ 1 2 x (ω + 2qπ)δ 1 2 y (ω + 2qπ)|ω + 2qπ| -H x -H y -d . ( 26
)
Proof The proof is divided into two parts. In a first part, we establish an asymptotic normality of quadratic variations of tangent fields of Z. Then, in a second part, we extend this result to quadratic variations of the field Z itself using a Slutsky lemma.

Part 1. The tangent field Zx of Z(x) at position x is given by Equation ( 10) of Theorem 1. We define a vector-valued increment fields Ṽ N of Zx whose terms at positions m ∈ Z d are given by

∀ (a, x) ∈ F × G, Ṽ N a,x [m] = k v[k] Zx p N x + m -T a k N . ( 27
)
Some properties of this field are stated in the following lemma.

Proposition 1 Assume that the field Ṽ N is obtained with a kernel of order K ≥ d 4 . Then, it is zero-mean, second-order stationary with an auto-covariance whose terms are defined, for any m, n ∈ Z d , by an integral of the form

Cov Ṽ N a,x [m], Ṽ N b,y [n] = 1 (2π) d [0,2π] d e i m-n,w hN a,x;b,y (ω)dω.
The function hN a,x;b,y is the multivariate spectral density of Ṽ N . It is given, for any

(a, x), (b, y) ∈ F × G, by hN a,x;b,y (w) = N -H x -H y ha,x;b,y (w)e i p N x -p N y ,w ,
where ha,x;b,y is defined by Equation [START_REF] Kolmogorov | Wienersche spiralen und einige andere interessante kurven im hibertschen raum[END_REF].

It is in L 2 ([0, 2π] d ).
In particular, for any (a, x) ∈ F × G, the variance terms of Ṽ N are

Var( Ṽ N a,x [m]) = ρa N 2H x C x (ϕ a ), (28)
where C x is defined by Equation [START_REF] Cohen | Fractional fields and applications[END_REF].

The proof of this proposition can be found in [START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF] (Supplementary materials, Lemma 1). For (a, x) ∈ F × G, let XN a,x be the normalized random field defined, for any m ∈ Z d , by

XN a,x [m] = N H x Ṽ N a,x [m] ρH x a C x (ϕ a )
.

According to Proposition 1, it is zero-mean, second order stationary with spectral density ha,x;a,x

ρH x a √ C x (ϕ a ) in L 2 ([0, 2π] d ). Now, let SN a,x = 1 √ n m∈V N ( XN a,x [m]) 2 -1 . ( 29
)
Applying a multivariate version of the Breuer Major Theorem established in [START_REF] Biermé | Central limit theorems and quadratic variations in terms of spectral density[END_REF] (Theorem 3.2), we obtain

( SN a,x ) a∈F f .d.d. -→ N →+∞ N (0, Σ (x) ), (30)
where the covariance matrix Σ (x) = (Σ a,x;b,x ) a,b∈F is defined with terms Σ a,x;b,x of Equation [START_REF] Istas | Quadratic variations and estimation of the local Hölder index of a Gaussian process[END_REF]. Now, to establish the convergence of the whole random vector ( SN a,x ) (a,x)∈F ×G , it suffices to show the convergence of the covariance terms Cov( SN a,x , SN b,y ). This is stated in the following lemma proved in Appendix A.

Lemma 1 Let SN be defined by Equation ( 29) and K > d 4 . Then

lim N →+∞ Cov( SN a,x , SN b,y ) = Σ a,x;b,y ,
where the term Σ a,x;b,y is given by Equation ( 25).

Part 2. Let now

S N a,x = 1 √ n m∈V N                N H x V N a,x [m] ρH x a C x (ϕ a )        2 -1         = √ n       N 2H x W N a,x ρ2H x a C x (ϕ a ) -1       ,
where W N a,x are quadratic variations associated to the field Z. We are going to establish the asymptotic normality

S N f .d.d.
-→

N →+∞ N (0, Σ). (31)
Assume for a moment that this convergence holds. Then, the asymptotic normality (24) of log-quadratic variations Y N can be obtained with the ∆method by applying the logarithm transform to

N 2H x W N a,x ρ2H x a C x (ϕ a )
and observing that log

      N 2H x W N a,x ρ2H x a C x (ϕ a )       = Y N a,x -ζ N a,x .
To establish the convergence (31), we use a Slutsky lemma. We observe that S N = SN + T N where, for (a, x) ∈ F × G,

T N a,x = 1 ρ2H x a C x (ϕ a ) 1 √ n m∈V N N 2H x (V N a,x [m]) 2 -N 2H x ( Ṽ N a,x [m]) 2 .
The convergence in distribution of SN was shown in Part 1. Hence, it suffices to show that T N converges to 0 in probability.

Let > 0. Using Markov and triangular inequalities, we get

P(|T N | ≥ ) ≤ c n a,x m∈V N N 4H x E N a,x [m], with c = -2 max a,x ( ρ2H x a C x (ϕ a )) -2
, and

E N a,x [m] = E (V N a,x [m]) 2 -( Ṽ N a,x [m]) 2 2 .
Furthermore, due to moment properties of Gaussian variables,

E N a,x [m] = 2 (E((V N a,x [m]) 2 ) 2 + (E( Ṽ N a,x [m]) 2 ) 2 -2(E(V N a,x [m] Ṽ N a,x [m])) 2 .
We conclude the proof using the following lemma proved in Appendix A.

Lemma 2 Take the same conditions as in Theorem 2. There exists a sequence (ζ N ) N which tends to 0 as N tends to +∞, such that, for large N

|N 2H x E((V N a,x [m]) 2 ) - ρ2H x a C x (ϕ a )| ≤ ζ N , ( 32
)
and |N 2H x E(V N a,x [m] Ṽ N a,x [m]) - ρ2H x a C x (ϕ a )| ≤ ζ N , ( 33
) hold for any m ∈ V N , (a, x) ∈ F × G.
Indeed, due to this lemma and Equation [START_REF] Matheron | The intrinsic random functions and their applications[END_REF],

N 4H x |E N a,x [m]| ≤ cζ 2 N for some c > 0, and any m ∈ V N , (a, x) ∈ F × G. Hence, P(|T N | ≥ ) ≤ c ζ 2
N for some c > 0. Therefore, T N converges to 0 in probability, and Assertion (31) holds.

Tests of heterogeneity

Setting

In this section, we aim at testing whether an observed AMFBF is spatially homogeneous or not. With regard to its definition in Section 2.2, an AMFBF will be considered as heterogeneous whenever the topothesy functions τ x or the Hurst functions β x characterizing its density vary with the position x.

Spatial heterogeneities of fields may appear in their local regularity, their directional properties, or both. The regularity of a field is heterogeneous whenever the Hurst index H x (see Equation ( 5)) is not constant. Its directional properties are heterogeneous when, from a position to another, the topothesy or Hurst functions change. Field heterogeneities can be further analyzed through the function B N

x defined by Equation ( 23). In the case when

H x is constant, variations of B N
x reveal some directional heterogeneities. In the general case, they may also reflect a regularity heterogeneity as B N x depends on H x .

Let O = {arg(u), u ∈ F } be the set of rotation angles of all transforms of F . We express the presence of an heterogeneity at positions of G in terms of two hypotheses:

H (1) 1 : ∃ x, y ∈ G, H x H y . H (2) 1 : ∃ ϕ ∈ O, ∃ x, y ∈ G, H x H y or B N x (ϕ) B N y (ϕ).
The hypothesis H

1 accounts for a regularity heterogeneity, and H

1 for an heterogeneity concerning either the regularity or directional properties. When the field regularity is presumed constant (ie ∀x ∈ G,

H x = H), H (2)
1 may only account for a directional heterogeneity. The negation of these hypotheses leads to two homogeneity hypotheses :

H (1) 0 : ∀ x ∈ G, H x = H. ( 34
)
H (2) 0 : ∀ ϕ ∈ O, ∀ x ∈ G, H x = H and B N x (ϕ) = B(ϕ).
In what follows, we develop statistical tests to validate H (j) 0 against H (j) 1

for j = 1, 2.

Fisher tests

The construction of statistical tests is grounded in Theorem 2. This theorem implies that log-variations Y N a,x are linked to log-scales ρ a by a linear model of the form

∀a ∈ F , x ∈ G, Y N a,x = ρ a H x + B N x (ϕ a ) + ε a,x , (35)
where the distribution of the vector (ε a,x ) (a,x)∈F ×G is close to the one of a centered Gaussian vector with covariance matrix Σ. In this relationship, the variables of interest H x and B N x (ϕ a ) appear as model parameters. Their values can be estimated and tested using classical techniques of generalized Gaussian linear models.

In particular, we can design some Fisher tests to check H (j) 0 against H (j)

1 for j = 1, 2. To define this test, we first express the model in a matricial form. For that, we create a multi-index (i, j, k

), i ∈ [[1, I]] referring to a ith position x i in G, j ∈ [[1, J]] to a jth rotation angle ϕ j of O, and k ∈ [[1, K j ]
] to a rescaling factor ρ jk of a kth transform of G, say a jk , among those with a rotation of angle ϕ j . Using this multi-index, we define two column random vectors Y = (Y N a jk ,x i ) ijk and = (ε a jk ,x i ) ijk of size IK, where

K = K 1 + • • • + K J is the cardinality of F .
We also form a parameter vector θ of size I(J + 1) whose terms are defined as

θ ml = H x m if l = 1, B N x m (ϕ l-1 ) if l > 1. using a multi-index (m, l) in [[1, I]] × [[1, J + 1]].
We eventually set a design matrix X of size IK × I(J + 1) having terms:

X ijk,ml =          ρ jk if m = i and l = 1, 1 if m = i and l = j + 1, 0 otherwise.
With these notations, Model ( 35) is equivalent to

Y = Xθ + , ∼ N (0, Σ). (36)
In this model, the parameter θ belongs to a vectorial space Θ = ((0, 1) × R J ) I . When one of the assumptions H (l) 0 given by Equations ( 34) holds, θ belongs to a subspace Θ (l) 0 of Θ of the form

Θ (l) 0 = {θ ∈ Θ, L (l) θ = 0},
for some matrix L (l) . For l = 1, L (1) is the matrix of size (I -1) × I(J + 1) whose terms are (2) is the matrix of size (I -1)(J + 1) × I(J + 1) whose terms are

L (1) m,ij =          1 if i = 1 and j = 1, -1 if i = m + 1 and j = 1, 0 otherwise. for m ∈ [[1, I -1]] and (i, j) ∈ [[1, I]] × [[1, J + 1]]. Likewise, for l = 2, L
L (2) mn,ij =          1 if i = 1 and j = n, -1 if i = m + 1 and j = n, 0 otherwise. for (m, n) ∈ [[1, I -1]] × [[1, J + 1]] and (i, j) ∈ [[1, I]]× ∈ [[1, J + 1]]. Let l = 1, 2.
The construction of a log-likelihood ratio test for the hypothesis H (l) 0 leads to a so-called Fisher statistics

F (l) = I(K -J -1) I(J + 1) -rank(L (l) ) θ L (l) (L (l) (X Σ -1 X) -1 L (l) ) -1 L (l) θ (Y -X θ) Σ -1 (Y -X θ) , ( 37
)
where θ is the generalized least square estimate of θ in Model (36

) θN = (X Σ -1 X) -1 X Σ -1 Y N . ( 38
)
Setting l = 1, 2, the decision of rejecting (resp. accepting) the null hypothesis

H (l) 0 is taken if F (l) ≥ s α (resp. ≤ s α ).
Under the null hypothesis, the statistic F (l) has a Fisher law distribution with degrees of freedom (I(J +1)-rank(L (l) ), I(K -(J + 1))). Given α ∈ (0, 1), the rejection threshold s α can be set accordingly so as to ensure that P

H (l) 0 (F (l) ≥ s α ) ≤ α,
meaning that the probability of wrong rejection remains below α.

To implement this test, we simply use an ordinary least square estimate of θ and replace Σ by an empirical covariance matrix computed on neighborhoods of positions of G.

Numerical study

So as to evaluate the test procedure presented in Section 4, we made some experiments on synthetic data. This data was simulated using the method developed in [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian field[END_REF].

This method is based on the principles of the so-called turning-band method: given a set of orientations

Θ = {θ i , i = [[1, n]]} in R 2 and a set of positive weights Λ = {λ i , i = [[1, n]]}, the random field Z to be simulated is approximated by an appropriate combination of independent random processes Y i Z Θ,Λ (x) = n i=1 λ i Y i ( u(θ i ), x ),
where u(θ i ) stands for a unit vector in the direction θ i . Using this approximation, the simulation of the field Z reduces to the simulation of n random processes Y i . In [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian field[END_REF], we specified the turning-band method for the simulation of AFBF. In particular, we showed that the band processes Y i should be a fractional Brownian motions with a Hurst index corresponding to the value of the Hurst function of the AFBF in the direction θ i . We also designed a dynamic programming algorithm to select some optimal directions θ i where processes Y i could be exactly and quickly simulated using a circulant embedding method.

Here, we propose to extend the application of this method to AMFBF in an algorithmic way. Given an AMFBF Z and a set of simulation points {x k , k = [[1, K]]}, a realization of Z at a point x k is obtained by simulating at position x k the tangent field Zx k of Z. Each tangent field is generated using a same pseudo-number sequence so as to maintain a coherent realization of Z accross different simulation points. This approach has not received any theoretical background, but was sufficient to test our method in the context of the paper.

Using this approach, we simulated AMFBF on a square grid of size 768× 728. We divided this grid into three regions: two regions of size 256×768 at the top and bottom of the grid, [ [START_REF] Ayache | A central limit theorem for the quadratic variations of the step fractional Brownian motion[END_REF]257]] × [ [START_REF] Ayache | A central limit theorem for the quadratic variations of the step fractional Brownian motion[END_REF]768]] and [[512, 768]] × [ [START_REF] Ayache | A central limit theorem for the quadratic variations of the step fractional Brownian motion[END_REF]768]], respectively, and the region in between. To apply the test procedure, we focused on the top and bottom regions. In each of these regions, the simulated AMFBF had a same tangent field.

Tangent fields of these regions were defined using two topothesy functions τ 0 and τ 1 , two constant Hurst functions η 0 ≡ H 0 and η 1 ≡ H 1 , and a translation parameter δ. For each experiment, these parameters were randomly and independently set: Hurst indices H 0 and H 1 were sampled from uniform distributions on (0.05, 0.95) and (H 0 , 0.95), respectively. The translation parameter was sampled from a uniform distribution in (0, π). We expanded topothesy functions jn | so as to ensure the positivity of functions. The first tangent field had parameters (τ 0 , η 0 ), the second (τ 0 , η 1 ), the third (τ 0 (• + δ), η 0 ), and the fourth (τ 1 , η 0 ).

We considered four AMFBF that were tangent to the first AFBF in the top region and to one of the four AFBF in the bottom region: AMFBF 1 to 4, the number referring to the tangent AMBF on the bottom region. The AMFBF 1 was actually homogeneous while AMFBF 2, 3, 4 accounted for heterogeneous fields with variations of regularity, orientations, and anisotropy forms, respectively. By construction, AMFBF 1, 3, and 4 were homogeneous in terms of regularity, and fulfilled Hypothesis H

(1) 0 of Equation [START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF]. By contrast, only AMFBF 1 satisfied Hypothesis H

(2) 0 . We made 5000 experiments, each containing realizations of the four AMFBF. To each realizations, we applied the test procedure to the check the homogeneity of the top and bottom regions. We analyzed both regions with quadratic variations using the filter v of order 1 defined by Equation ( 13) with (L 1 , L 2 ) = (0, 2). These variations were computed for all transformations T u associated to u ∈ Z 2 , √ 2 ≤ |u| ≤ 4.3. Hence, we had rotations of angles 0 (u ∝ (1, 0)), π/4 (u ∝ (1, 1)), π/2 (u ∝ (0, 1)) or 3π/4 (u ∝ (1, -1)), and rescalings of factors |u| between √ 2 and 4.3. In particular, we had three quadratic variations in each direction.

0 against H

0 against H

Fig. 2: Empirical risks of error for the two heterogeneity tests.

Next, we computed Fisher statistics F (1) and F (2) for testing hypotheses H

(1) 0 and H

(2) 0 of Equations [START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF], respectively. For each test and type of sim-ulated AMFBF, we assessed rates of wrong decisions taken over the dataset for different values of rejection threshold s α . In the case when the AMFBF fulfilled H (u) 0 , such empirical rates gave us estimates of the risk of error of type I (error made when rejecting H (u) 0 ) whereas, in the opposite case, they were estimates of the risk of error of type II (error made when rejecting H (u) 1 ). In Figure 2, these estimated risks are plotted in percent as a function of the theoretical risk of error of type I associated to the threshold s α .

For both tests, we notice that red lines representing estimated risks of error of type I with respect to the theoretical ones were close to the identity line. This means that empirical distributions of Fisher statistics under the null hypotheses approximately matched the theoretical ones. It also suggests that our simulation and estimation procedures were well-suited to the theoretical model.

Besides, we observe that estimated risks of error of type II depended on the heterogeneity nature of simulated AMFBF. The regularity heterogeneity of AMFBF 2 was better detected than directional heterogeneities of AMFBF 3 and 4. For AMFBF 2, the risk of missing a regularity heterogeneity was about 3.5% (with the first test) and 0.5% (with the second test) while the risk of error of type I was only about 0.05%. By contrast, for AMFBF 3 and 4, the risk of missing a directional heterogeneity with the second test was above 15% for the same level of risk of error of type I. Heterogeneities resulting from a change of the topothesy orientation (AMFBF 3) was particularly difficult to detect. When the risk of error of type I was about 1%, the risk of missing such heterogeneities was about 10%.

Discussion

We proposed statistical tests to decide whether an observed random field is heterogeneous or not, the heterogeneity resulting from spatial variations of the regularity of the field, its directional properties, or both. The test procedure was based on quadratic variations specifically designed for a local and oriented analysis of the observed field. For developing the theory, we set a framework of anisotropic multifractional Brownian field heterogeneity. In this framework, we established the asymptotic normality of quadratic variations. This result highlighted a Gaussian linear relationship between quadratic variations and parameters related to the regularity and directional features of the model. From this relationship, we designed our tests using Fisher statistics of linear Gaussian models. We evaluated the test procedure using data simulated with a turning-band method. Results showed the feasability of the procedure: applied to datasets mixing realizations of isotropic and anisotropic fields, it could correctly detect anisotropic fields with error rates ranging from 0.5 % to 10 % while maintaining a low error rate (below 1%) of detecting isotropic fields. Errors of detecting anisotropic fields varied depending on the type of heterogeneity. Regularity heterogeneities were the easiest to detect while those due to orientation changes were the most difficult.

Random fields derived from the fractional Brownian motion has proved to be useful models for the analysis of rough textures in the domain of image processing. In particular, anisotropic fractional Brownian fields could be applied to analyze textures appearing in images as various as bone radiographs [START_REF] Biermé | Parametric estimation for Gaussian operatorscaling random fields and anisotropy analysis of bone radiograph textures[END_REF], mammographs [START_REF] Biermé | Analysis of texture anisotropy based on some Gaussian fields with spectral density[END_REF][START_REF] Richard | Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography[END_REF][START_REF] Richard | Analysis of anisotropic Brownian textures and application to lesion detection in mammograms[END_REF][START_REF] Richard | Some anisotropy indices for the characterization of Brownian textures and their application to breast images[END_REF] , or photographic films [START_REF] Richard | Anisotropy of Hölder Gaussian random fields: characterization, estimation, and application to image textures[END_REF]. Accounting for directional properties of textures, they enabled to achieve various image classification tasks. However, as they applied homogeneous textures, they were only suitable for global processing tasks. The introduction of an anisotropic heterogeneous Brownian model enables to overcome this limitation and widen the scope of applications of Brownian models. It notably provides practionners with a model which may account for the presence of several textures within a single image. It also sets a theoretical framework for image segmentation, task which consists of delimiting regions where these different textures are. In such a task, heterogeneity tests can be used to decide if textures contained in different regions are similar or not.

In this paper, the analysis of the field was focused on testing its heterogeneity. But, it could also be used to characterize the field heterogeneity. Following an approach that is similar to the one developed in [START_REF] Richard | Analysis of anisotropic Brownian textures and application to lesion detection in mammograms[END_REF][START_REF] Richard | Some anisotropy indices for the characterization of Brownian textures and their application to breast images[END_REF][START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF] for AFBF, estimates involved in our field analysis could serve for defining some measures of the heterogeneity of AMFBF. Moreover, the approach proposed in [START_REF] Richard | Anisotropy of Hölder Gaussian random fields: characterization, estimation, and application to image textures[END_REF] to estimate the so-called asymptotic topothesy of AFBF could also be extended to the one of tangent fields of AMFBF. This would path the way to the local estimation of the whole topothesy and Hurst functions of AMFBF.

A Proof of supporting lemmas and propositions

We first show a main technical lemma.

Lemma 3 (Main lemma) Let Z be an anisotropic multifractional Brownian field which fulfills all assumptions of Section 2.2. Consider an increment kernel v of order K ≥ 0 with a support [[0, L]] for some L ∈ N d , as well as two transforms T 1 and T 2 composed of a rotation and a rescaling. Fix a position x ∈ R d . Choose a positive sequence (ε t ) t>0 such that

lim t→0 + | log(t)|ε η t = 0. ( 39
)
For any t > 0, set the local neighborhood of x

V t = {y ∈ R d , |y -tT j k -x| ≤ ε t , ∀ k ∈ [[0, L]], j = 1, 2}.
and, for y ∈ R d and j ∈ {1, 2}, denote

V tj (y) = k∈[[0,L]] v[k]Z(y -tT j k).
For any y ∈ R d , define

I(y) = R d v(T 1 w) v(T 2 w)e i y,w δ x (w)|w| -2H x -d dw, ( 40
)
where H x and δ x are defined by Equations ( 5) and ( 7), respectively. Then there exists a positive sequence (γ t ) t>0 converging to 0 as t → 0 + such that, for any small t > 0,

∀y 1 , y 2 ∈ V t , t -2H x E (V t1 (y 1 )V t2 (y 2 )) -I(t -1 (y 1 -y 2 )) ≤ γ t . ( 41
)
We also state a variant of this lemma whose proof can be established in a same way.

Lemma 4 (A variant of the main lemma) Take the same conditions as in Lemma 3. Set x ∈ R d . Let Zx be the tangent field of Z at x defined by Equation [START_REF] Biermé | Parametric estimation for Gaussian operatorscaling random fields and anisotropy analysis of bone radiograph textures[END_REF].

For y ∈ R d , let V t1 (y) = k∈[[0,L]] v[k]Z(y -tT 1 k), and V t2 (y) = k∈[[0,L]] v[k] Zx (y -tT 2 k).
Then there exists a positive sequence (γ t ) t>0 converging to 0 as t → 0 + such that, for any small t > 0, Equation (41) holds for the integral I defined by Equation (40).

Proof (Lemma 3)

We first show the result in the case of an elementary kernel v of order 0. Then, we extend it to the case of a general kernel of order K ≥ 0.

So, let us begin by assuming that, for some andv[k] = 0 for all other k ∈ Z d (e i denotes the unit vector of R d whose ith components is 1 and others are 0). Then, for t > 0 and y 1 , y 2 ∈ V t ,

i ∈ [[1, d]], v[0] = 1, v[e i ] = -1,
E (V t1 (y 1 )V t2 (y 2 )) = |w|≥A R t1 (w)R t2 (w)dw, (42)
where, for j ∈ {1, 2}, R tj (w) = (e i y j ,w -1) f y j (w) -(e i y j -t j ,w -1) f y j -t j (w), with t j = tT j e i .

Specifying f x with its expression (1), we can further write R tj (w) = 3 k=1 R tjk (w) with

R tj1 (w) = e i y j ,w (1 -e -i t j ,w ) τ y j -t j (w) |w| -β y j -t j (w)-d 2 , R tj2 (w) = (e i y j ,w -1) τ y j (w) -τ y j -t j (w) |w| -β y j (w)-d 2 , R tj3 (w) = (e i y j ,w -1) τ y j -t j (w) |w| -β y j (w)-d 2 -|w| -β y j -t j (w)-d 2 .
(43)

Now, let

D t = t -2H x E(V t1 (y 1 )V t2 (y 2 )) -I(t -1 (y 1 -y 2 )). ( 44
)
After a variable change in the integral I, we get

D t = t -2H x (D t1 -D t2 + D t3 )
where

D t1 = |ω|>A R t11 (w) R t21 (w) -e i y 1 -y 2 ,w (e -i t 1 ,w -1)(e i t 2 ,w -1)δ x (w)|w| -2H x -d dω, D t2 = |ω|≤A e i y 1 -y 2 ,w (e -i t 1 ,w -1)(e i t 2 ,w -1) δ x (w)|w| -2H x -d dω, D t3 = (k,l) (1,1) |w|>A R t1k (w) R t2l (w)dw.
Using

|e iu -1| = 2| sin( u 2 )| = O u→0
(|u|), we obtain that

|D t2 | ≤ ct 2 |w|≤A |w| 2-2H x -d dw,
where c > 0 and the integral on the right-hand side is finite since 0 < H x < 1. Thus,

t -2H x D t2 = O t→0 + (t 2(1-H x ) ). (45)
Due to the Cauchy-Schwarz inequality, D t3 ≤ (k,l) (1,1) ∆ t1k ∆ t2l , where

∆ tjk = |w|≥A |R tjk (w)| 2 d. ( 46
)
To deal with D t3 , we establish the following lemma.

Lemma 5

We have the following inequalities.

t -2H x ∆ tj1 ≤ c max(|T j | 2H , |T j | 2 β ), (47)
t -2H x ∆ tj2 = |T j | 2α O t→0 + (t 2α-2H x ), (48)
t -2H x ∆ tj3 = |T j | 2η O t→0 + (t 2η-2H x ). (49)
In polar coordinates,

∆ tj1 = S +∞ A |1 -e -iρ t j ,s | 2 τ y j -t j (s)ρ -2β y j -t j (s)-1
dρds.

Since τ is bounded and, for all w, 0 < H ≤ H x ≤ β x (w) ≤ β, a variable change r = tT j ρ in ∆ tj1 yields

∆ tj1 ≤ c max(|T j | 2H , |T j | 2 β ) S ( r<1 |1 -e -ir e j ,s | 2 r -2 β-1 dr + r>1 r -2H-1 dr)J 0 (s)ds
where

J 0 (s) = S t 2β y j -t j (s)-2H x ds,
for some c > 0. In the right-hand side of the inequality, both integrals over r are finite. Hence,

∆ tj1 ≤ c max(|T j | 2H , |T j | 2 β ) S J 0 (s)ds.
Observe now that

t 2β y j -t j (s)-2H x = t 2β x (s)-2H x t 2β y j -t j (s)-2β x (s) = t 2β x (s)-2H x exp log(t)[2β y j -t j (s) -2β x (s)] .
Using Assumption [START_REF] Benassi | Elliptic Gaussian random processes[END_REF],

|2β y j -t j (s) -2β x (s)| ≤ 2 η t .
Due to Assumption (39), we then deduce the limit:

lim t→0 + | log(t)|[2β x 0 +tv (s) -2β x 0 (s)] = 0. (50)
Consequently, lim

t→0 + t 2β y j -t j (s)-2β x (s)
= 1. Therefore, we get

J 0 (s) ≤ c S t 2β x (s)-2H x ds, since β x (s) ≥ H x , ∀s ∈ S, J 0 (s) is finite. Hence, Equation (47) holds.
In polar coordinates,

∆ tj2 = S +∞ A ρ -2β y j (s)-1
|1e iρ y j ,s | 2 τ y j (s)τ y j -t j (s) 2 dρ ds.

Using the variable change r = ρ y j , s , we obtain

∆ tj2 ≤ c +∞ 0 |1 -e -ir | 2 r -2β y j (s)-1 S τ y j (s) -τ y j -t j (s) 2 ds dr.
Using Assumption (9), we get

∆ tj2 ≤ c |T j | 2α t 2α 1 0 |1 -e -ir | 2 r -2 β-1 dρ + +∞ 1 r -2H-1 dr ,
for small t and some c > 0. The integrals on the right-hand side are finite. Hence, Equation ( 48) holds.

Applying a mean value inequality to the function u → ρ u , we obtain that

|ρ -β y j (s) -ρ -β y j -t j (s) | ≤ c|β y j (s) -β y j -t j (s)|| log(ρ)| max(ρ -H , ρ -β ).
Hence, using Assumption (8), we get

|ρ -β y j (s) -ρ -β y j -t j (s) | ≤ c |T j | η t η | log(ρ)| max(ρ -H , ρ -β ),
for small t and some c > 0. Therefore, in polar coordinates, we have

∆ tj3 ≤ c |T j | 2η t 2η 1 0 |1 -e iρ y j ,s | 2 ρ -2 β-1 | log(ρ)|dρ + +∞ 1 ρ -2H-1 | log(ρ)|dρ .
Since, for any u > 0, | log(t)|t u (resp. | log(t)|t -u ) tends to 0 as t tends to 0 (resp. +∞), the integrals on the right-hand side are bounded by a constant proportional to a sum

ν 0 ρ 2-2( β+ )-1 dρ + 1 ν ρ -2 β dρ + B 1 ρ -2H-1 dρ + +∞ B ρ -2(H-2 )-1 dρ (51)
of finite integrals defined for some 0 < ν < 1, B > 1, and 0 < < min(H, 1 -β). Hence, Equation (49) holds.

Eventually, using Lemma 5, we get

t -2H x D t3 = O t→0 + (t 2(min(α,η)-H x ) ). (52)
Besides, D t1 ≤ D t11 + D t12 , where, in polar coordinates,

D t1j = +∞ A S |e -iρ t 1 ,s -1| |e iρ t 2 ,s -1|d t1j (ρ, s)dsdρ, d t1 (ρ, s) = δ x (s)ρ -2H x -1 ρ 2H x -β y 1 -t 1 (s)-β y 2 -t 2 (s) -1 , d t2 (ρ, s) = τ y 1 -t 1 (s) τ y 2 -t 2 (s) -δ x (s) ρ -β y 1 -t 1 (s)-β y 2 -t 2 (s)-1 .
Consider the set E x defined by Equation ( 6). Due to the definition of δ x in Equation ( 7), the integrand d t11 (s) vanishes for s ∈ E c x . Applying the mean value theorem to the function u → ρ u , and using Assumption [START_REF] Benassi | Elliptic Gaussian random processes[END_REF], we obtain that, for

s ∈ E x , |ρ 2H x -β x t k (s)-β x t l (s) -1| ≤ 2| log(ρ)| max(ρ -2ε η t , ρ 2ε η t )ε η t .
Moreover, due to Assumption (39), ε η t , t 2ε η t and t -2ε η t are bounded for small t. Therefore, max(ρ -2ε η t , ρ 2ε η t ) ≤ c max((tρ) -2 , (tρ) 2 ) for some < min(H, 1-β) and small t. On the other hand, | log(ρ)| ≤ 1 + log(tρ) | log(t)| for small t. Therefore, we have

d t1 (ρ, s) ≤ c(tρ) -2H x -1 max((tρ) -2 , (tρ) 2 ) 1 + log(tρ) t 2H x +1 | log(t)|ε η t .
Hence, after a variable change r = tρ in the integral, we get

D t11 ≤ O t→0 + t 2H x | log(t)|ε η t J 1 , where J 1 is equal to S +∞ 0 |e -iρ T 1 e i ,s -1||e iρ T 2 e i ,s -1| 1 + log(ρ) max ρ -2(H-)-1 , ρ -2( β+ )-1 dρds
, an integral bounded by a sum of finite integrals of the form (51). Hence,

t -2H x D t11 = O t→0 + | log(t)|ε η t .
Using Assumption [START_REF] Benassi | Elliptic Gaussian random processes[END_REF],

ρ -β x 1 -t 1 (s)-β x 2 -t 2 (s)-1 ≤ cρ -2β x (s)-1 max(ρ 2ε η t , ρ -2ε η t ).
Hence, due to Assumption (39),

ρ -β x 1 -t 1 (s)-β x 2 -t 2 (s)-1 ≤ ct 2β x (s)+1 max((tρ) -2( β+ )-1 , (tρ) -2(H-)-1 ),
for small t, and some < min(H, 1 -β). Consequently,

D t12 ≤ c S τ x 1 -t 1 (s) τ x 2 -t 2 (s) -δ x (s) t 2β x (s) J 2 (s)ds,
where the integrals

J 2 (s) = +∞ 0 |e -iρ T 1 e i ,s -1| |e iρ T 2 e i ,s -1| max(ρ -2(H-)-1 , ρ -2( β+ )-1 )dρ
are bounded independently of s by a sum of finite integrals of the form (51). Furthermore, for s ∈ E x c , δ x (s) = 0 and τ x is bounded. On the other hand, for x ∈ E x , δ x (s) = τ x (s), and τ x 1 -t 1 (s) τ x 2 -t 2 (s)δ x (s) can be written as 2 j=1 ( τ x j -t j (s)τ x (s)) + τ x (s) 2 j=1 ( τ x j -t j (s)τ x (s)).

Therefore, using a Cauchy-Schwarz inequality, we obtain t -2H x D t12 ≤ c(ζ t1 + ζ t2 ), with

ζ t1 = E c x t 2(β x (s)-H x ) ds,
and

ζ t2 = 2 j=1 S τ x j -t j (s) -τ x (s) 2 ds + 2 j=1 S τ x j -t j (s) -τ x (s) 2 ds.
From Assumption [START_REF] Benassi | Elliptic Gaussian random processes[END_REF], it follows that

ζ t2 = O t→0 + ε α t . Besides, let F x,t = s ∈ S, β x (s) -H x > | log(t)| -1 2 .
Then,

ζ t1 = F x,t t 2(β x (s)-H x ) ds + E c x \F x,t t 2(β x (s)-H x ) ds ≤ c       t 2| log(t)| -1 2 + µ(E c x \F x,t )       ,
where µ(E c x \F x,t ) is the measure of E c x \F x,t on S. This measure tends to 0 as t tends to 0, as well as the first term on the right-hand side of the inequality. As a consequence, there exists (ζ t ) t converging to 0 as t tends to 0 + such that t

-2H x D t12 = O t→0 + (max(ζ t , ε α t )
), and

t -2H x D t1 = O t→0 + (max(ζ t , ε α t , | log(t)|ε η t )). (53)
Using Equations (45), ( 52), (53), we finally establish Equation (41) for some sequence (γ t ) t satisfying

γ t = O t→0 + max ζ t , ε α t , | log(t)|ε η t , t 2(1-H x ) , t 2(min(α,η)-H x ) .
This sequence converges to 0 as t → 0 + , due to Assumption (39) and since β < min(α, η), and

H x < β < 1.
Let us now turn to the general case when v is a kernel of order K ≥ 0. In this case, the

characteristic polynomial Q v (z) = k∈[[0,L]] v[k]z k 1 1 • • • z k d d of the kernel v admits (1, • • • , 1) as a root.
Consequently, for some i ∈ [ [1, d]], there exists a polynomial Q ṽ , such that

Q v (z) = (1 -z i )Q ṽ (z).
By identification, one can check that the polynomial

Q ṽ = k∈[[0,L-e i ]] d ṽ[k]z k 1 1 • • • z k d d , where v[k] = ṽ[k] -ṽ[k -e i ] for all k ∈ [[e i , L -e i ]] d , v[0] = ṽ[0], and v[L] = -ṽ[L -e i ].
As a consequence, we may write

V tj (y) = k∈[[0,L-e i ]] d ṽ[k] V 0 tj (y -tT j k),
where

V 0 tj (y) = m∈[[0,e i ]] d v 0 [m]Z(y -tT j m
) is an increment of order 0 defined with the kernel v 0 having terms v 0

[0] = 1, v 0 [e i ] = -1. Moreover, v(w) = Q v (e iw 1 , • • • , e iw d ) = (1 -e i e i ,w )Q ṽ (e iw 1 , • • • , e iw d ), so that I(y) = k,l∈[[0,L-e i ]] d ṽ[k] ṽ[l]I 0 (y -T 1 k + T 2 l), with I 0 (u) = R d
(1e -i e i ,T 1 w )(1e i e i ,T 2 w )e i u,w δ x (w)|w| -2H x -d dw.

So, in this case, the quantity D t defined in Equation (42) may be expressed as

D t = k,l∈[[0,L-e i ]] d ṽ[k] ṽ[l]D 0 t [k, l], with D 0 t [k, l] = t -2H x E(V 0 t1 (y 1 -tT 1 k)V 0 t2 (y 2 -tT 2 l)) -I 0 (t -1 (y 1 -y 2 -tT 1 k + tT 2 l)). This implies that |D t | ≤ c k,l∈[[0,L-e i ]] d |D 0 t [k, l]|, for some c > 0. But, from the first part, each D 0 t [k, l]
satisfies Equation (41) for a sequence (γ t ) t . So, does D t .

Proof (Lemma 1) Let SN a,x be as given by Equation [START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF]. Using moment properties of Gaussian variables, we obtain

Cov( SN a,x , SN b,y ) = 1 n m,n∈V N Cov(( XN a,x [m]) 2 , ( XN b,y [n]) 2 ) = 1 (2N + 1) d m,n∈[[-N ,N ]] d 2 E( XN a,x [m] XN b,y [n]) 2 -1 = k∈[[-2N +1,2N +1]] d d 1 - |k i | 2N + 1 |r[k + p N x -p N y ]| 2 -1,
where

r[k] = [0,2π] d e i k,w ha,x;b,y (ω)dω. ρH x a C x (ϕ a ) ρH y b C y (ϕ b ) , (54)
with notations of Equations ( 23) and [START_REF] Kolmogorov | Wienersche spiralen und einige andere interessante kurven im hibertschen raum[END_REF]. The term r[k] is a Fourier coefficient of a function which is in L 2 ([0, 2π] d ) for K > d/4 (see [START_REF] Richard | Tests of isotropy for rough textures of trended images[END_REF]). Using the Bessel Parseval equality, we thus have Hence, to establish the proposition, it suffices to show that

S N = k∈[[-2N +1,2N +1]] d |r[k + p N x -p N y ]| 2 -|r[k]| 2
converges to 0 as N tends to +∞. If x = y, this sum vanishes and the convergence is trivial. So, let us assume that x y. Since |r[k]| tends to 0 as |k| tend to +∞, we have

|S N | ≤ c k∈[[-2N +1,2N +1]] d s N [k], (55)
with s N [k] = |r[k + p N x -p N y ] -r[k]| and c > 0. Moreover, s N [k] ≤ c [0,2π] d (e i k+p N x -p N y ,ω -e i k,ω ) ha,x;b,y (ω)dω ≤ c [0,2π] d e -i p N y -p N x 2
,ω e i k,ω sin( p N xp N y , ω /2) ha,x;b,y (ω)dω , for some c > 0. Therefore,

s N [k] ≤ c R d e -i p N y -p N x 2 -k,ω b(ω)dω , ( 56
) where b(ω) = | sin( p N x -p N y , ω /2)| |ω| -H x -H y -d . Let us now study the integral B(u) = R d e -i u,ω b(ω)dω.
Assume that all components of u are non-zero. Using integration by parts on the first component of w, we get that B(u) is equal to

R d-1 m 1 e i u m ,ω m          e i u 1 ,ω 1 -1 iu 1 b(ω) +∞ -∞ - +∞ -∞ (e i u 1 ,ω 1 -1) iu 1 ∂b ∂ω 1 (ω)dω 1          m 1 dω m . Since lim |w|→+∞ b(w) = 0, it follows that B(u) = - 1 iu 1 R d m 1 e i u m ,ω m (e i u 1 ,ω 1 -1) ∂b ∂ω 1 (ω)dω.
Repeating the process on other components of w, we obtain

B(u) = i d m∈[[1,d]] 1 u m R d m∈[[1,d]]
(e i u l ,ω l -1)

∂ d b ∂ω 1 • • • ∂ω d (ω)dω,
so that, for some c > 0,

|B(u)| ≤ c m∈[[1,d]] 1 |u m | .
Hence, using Equation (56), we eventually get

s N [k] ≤ c m J 1 |p N x,m -p N y,m | m∈J 1 |k m | , where J = {m ∈ [[1, d]], x m -y m = 0} is a non-empty set of indices.
Using Condition [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF], it follows that

s N [k] = O N →+∞ (N -(d-|J|) ) m∈J 1 |k m | ,
where |J| stands for the cardinal of J. Moreover, we have

k m ∈[[-2N +1,2N +1]] |k m | -1 ≤ 1 + 2 2N 1 1 x dx ≤ 1 + 2 log(2N ).
Therefore, from Equation (55), we obtain

|S N | ≤ (1 + 2 log(2N )) |J| N d-|J| O N →+∞ (N -(d-|J|) ). Due to Condition (19), N = O N →+∞ (N N ). Thus, |S N | = O N →+∞ (1 + log(N ) + log( N )) |J| d-|J| N . If |J| = 0 then |S N | = O N →+∞ ( d N ). If |J| > 0, then |S N | = O N →+∞ (1 + log(N ) + log( N )) d/|J|-1 N ) |J| .
Hence, in both cases, lim N →+∞

|S N | = 0 due to Conditions [START_REF] Clausel | Explicit construction of operator scaling Gaussian random fields[END_REF].

Proof (Lemma 2) Since Z is the sum of two independent fields Z L and Z H , E((V N a,x [m]) 2 ) can be decomposed as a sum of E((V N L,a,x [m]) 2 ) and E((V N H,a,x [m]) 2 ) where V N L,a,x [m] and V N H,a,x [m] are increment fields of Z L and Z H , respectively. Since the increment is of order K ≥ M, we have

E((V N L,a,x [m]) 2 ) = R d v T a w N 2 g(w)dw.
We split the integral above into the sum of integrals on two domains, {w, |w| > A} and {w, |w| < A}.

Since v is of order K, we have

| v(u)| 2 = O u→0 + (|u| 2K+2
). Thus, for N large enough,

E N 1 = |w|<A v T a w N 2 g(w)dw ≤ N -2K-2 |w|<A |w| 2K+2 g(w)dw.
Since Z L is an intrinsic random field of order M and K ≥ M, the integral on the right-hand side is finite and

E N 1 = O N →+∞ (N -2K-2
). Using Assumption [START_REF] Biermé | Central limit theorems and quadratic variations in terms of spectral density[END_REF], we also have

E N 2 ≤ c |w|>A v T a w N 2 |w| -2 H-d dw.
After a variable change ζ = w/N , we obtain

E N 2 = O N →+∞ (N -2 H ) R d | v(ζ)| 2 |ζ| -2 H-d dζ.
Since the integral on the right-hand side is finite, we get

E N 2 = O N →+∞
(N -2 H ). Therefore,

N 2H x E((V N L,a,x [m]) 2 ) = O N →+∞ (N 2(H x -H) ) (57)
and converges to 0 as N tends to +∞ since H > β ≥ H x .

To establish the convergence of N 2H x E((V N H,a,x [m]) 2 ), we use Lemma 3. Indeed, set t = 1/N and ε t = N . Then, due to Conditions (20) (i) and (ii), ε t fulfills Condition (39). Setting y 1 = y 2 = p N

x +m N for an arbitrary m ∈ V N , we observe that, for j = 1, 2, V tj (y j ) =

V N H,a,x [m] when T j = T a . We further notice that I(0) = ρ2H x a C x (ϕ a ). Hence, Lemma 3 implies that, for any a ∈ F , m ∈ V N ,

|N 2H x E((V N H,a,x [m]) 2 ) - ρ2H x a C x (ϕ a )| ≤ γ 1 N , ( 58
)
for some sequence (γ t ) t converging to 0 as t tends to 0 + . Equation [START_REF] Richard | Analysis of anisotropic Brownian textures and application to lesion detection in mammograms[END_REF] follows from Equation (N 2(H x -H) ). Equation ( 33) can be established in a same way using the following lemma.

B Proof of Theorem 1

Proof Let x ∈ R d . We aim at proving that the random field Z is Locally asymptotically selfsimilar of order H x at x, and admits the AFBF Zx of Equation ( 10) as a tangent field. For t > 0, let V x,t be the Gaussian field defined, for any u ∈ R d , by

V x,t (u) = Z(x + tu) -Z(x) t H x . ( 59
)
We shall show that, for any u ∈ R d , {V x,t (u)} u∈R d converges to { Zx (u)} u∈R d , as t → 0, in distribution on the space of continuous functions endowed with the topology of the uniform convergence on compact sets [START_REF] Falconer | Tangent fields and the local structure of random fields[END_REF].

For that, we use a convergence criterion which is stated in Theorem 2.1.10 of [START_REF] Cohen | Fractional fields and applications[END_REF]. Due to this criterion, we only have to show 1. that V x,t converges to Zx in terms of finite dimensional distributions, as t → 0, 2. and that the sequence of random fields (V for some positive constant C 0 (I) which may depend on I, and positive constants α 1 , α 2 .

Convergence in finite dimensional distribution. Due to the Lévy theorem, the convergence of finite distributions of a field can be studied using characteristic functions of field variables. In the case when the field is Gaussian, it can be further established through the field autocovariance which determines characteristic functions. Here, since fields are Gaussian, it suffices to show that, for any u, v ∈ R d , lim t→0 E(V x,t (u)V x,t (v)) = E( Zx (u) Zx (v)).

(61)

For that, we apply Lemma 3 of Appendix A to variations V t1 (x) and V t2 (x) defined with a kernel v having terms v[0] = 1, v[e 1 ] = -1, v[k] = 0, otherwise, and transforms

T 1 = u 1 -u 2 u 2 u 1 and T 2 = v 1 -v 2 v 2 v
D t = E (V x,t (u) -V x,t (v)) 2 = E (Z(x + tu) -Z(x + tv)) 2 |t| 2H x . ( 63
)
Notice that Z(x + tu) -Z(x + tv) =

T = u 1 -v 1 -u 2 + v 2 u 2 -v 2 u 1 -v 1 .
Hence, setting V t1 (x + tu) = V t2 (x + tu) = Z(x + tu) -Z(x + tv) in Equation (42), we obtain

D t = t -2H x |ω|≥A |R t (ω)| 2 dω, ( 64
)
where R t (ω) = (e i x+tu,ω -1) f x+tu (ω)-(e -i x+tv,ω -1) f x+tv (ω). Furthermore, R t (ω) = 3 k=1 R tk (ω), where terms R tk (ω) are defined as in Equations (43). Therefore,

D t = t -2H x

Fig. 1 :

 1 Fig. 1: Realizations of different multifractional Brownian fields whose spatial properties vary along the vertical direction: (a) an isotropic one with a varying regularity, (b) an anisotropic one with varying pattern orientations, and (c) another anisotropic one with a varying degree of anisotropy.

  jn sin(2s), and, for j ∈ [[START_REF] Ayache | A central limit theorem for the quadratic variations of the step fractional Brownian motion[END_REF][START_REF] Clausel | Explicit construction of operator scaling Gaussian random fields[END_REF]] and u = 1, 2, sampled coefficients α (u) jn independently from centered normal distributions with a variance of 1/(1+n 2 ); we set α 0

  type I (with AMFBF 1) of type II (with AMFBF 2) of type I (with AMFBF 3) of type I (with AMFBF 4) (a) Test of H

  (with AMFBF 1) of type II (with AMFBF 2) of type II (with AMFBF 3) of type II (with AMFBF 4) (b) Test of H

  2π] d | ha,x;b,y (ω)| 2 dω. ρ2H x a C x (ϕ a ) ρ2H y b C y (ϕ b ) = Σ a,x;b,y .

1 N

 1 (57) and (58) for ζ N = γ

  k v[k]Z(x + tu -tT k), for the filter v having terms v[0] = 1, v[1, 0, • • • , 0] = -1, v[k] = 0 otherwise,and the transform

3 k

 3 ,l=1 |w|>A R tk (w) R tl (w)dw.Consequently, due to the Cauchy-Schwarz inequality, D t ≤ t -2H x 3 k,l=1 ∆ tk ∆ tl , where ∆ tk are as in Equation (46). Hence, Lemma 5 yieldsD t ≤ c max |T | 2 β , |T | 2H + (|T | 2η + |T | 2α ) O t→0 + (t 2(min(α,η)-H x ) ) + max |T | β+α + |T | β+η , |T | H+α + |T | H+η O t→0 + (t min(α,η)-H x ) + |T | α+η O t→0 + (t α+η-2H x ).Noticing that |T | = |u -v| and using assumptions that H < β < α < 1 and H < β < η < 1, we further haveD t ≤ c max |u -v| 2 β , |u -v| 2H 1 + max |u -v| min(α,η)-H , |u -v| 2(max(α,η)-β) ,the second term reaching a bound C 2 (I) for any (u, v) on the compact set [-I, I] 2 × [-I, I] 2 . Therefore,sup t>0 E (V x,t (u) -V x,t (v)) 2 ≤ C 2 (I) max |u -v| 2 β , |u -v| 2H . (65)Consequently, choosing p ∈ N so that pH > d, we obtainE (V x,t (u) -V x,t (v)) 2p ≤ C p (I) max |u -v| 2p β , |u -v| 2pH . (66)In other words, the tightness property (60) is shown for α 1 = 2p and α 2 = 2pHd.

  x,t n ) n∈N * , defined with t n = 1/n ∈ [0, 1], satisfies a tightness property∀I > 0, ∀u, v ∈ [-I, I], sup n>0 E |V x,t n (u) -V x,t n (v)| α 1 ≤ C 0 (I)|u -v| d+α 2 , (60)

  [START_REF] Ayache | A central limit theorem for the quadratic variations of the step fractional Brownian motion[END_REF] ,respectively. Let us notice that V t1 = t H x V x,t (u) and V t2 = t H x V x,t (v). Hence, setting t = t max (|u| ∞ , |v| ∞ ), we get lim d v(T 1 w) v(T 2 w)δ x (w)|w| -2H x -d dw, (62)the later integral corresponding to the term E( Zx (u) Zx (v)) of the autocovariance of Zx . Tightness property. Let us consider with t = 1 n ∈ [0, 1]

t→0 E(V x,t (u)V x,t (v)) = I(0) = R


PhD Marc Bitoun

MD Jorge Alfredo Bevilacqua

Bernard Prudhon

Svetlana Maugenre

MD Ana Lia Taratuto

MD)P Soledad Monges

MD)P Fabiana Lubieniecki

MD Claude Cances

MD Emmanuelle Uro-Coste

MD Michèle Mayer

Bernard Prudhonp

Svetlana Maugenrep

MD)P Michel Fardeau

MD Norma B Romero

PhD)P Pascale Guicheney
email: p.guicheney@myologie.chups.jussieu.fr

Dynamin 2 mutations cause sporadic centronuclear myopathy with neonatal onset

Keywords: Dynamin 2, Centronuclear Myopathy, Autosomal dominant inheritance. Neonatal onset

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

Introduction

The centronuclear myopathies 1 are a heterogeneous group of rare neuromuscular disorders including "Myotubular myopathy" for the severe X-linked recessive form (XLMTM) due to myotubularin mutations [START_REF] Laporte | A gene mutated in X-linked myotubular myopathy defines a new putative tyrosine phosphatase family conserved in yeast[END_REF] and "Centronuclear myopathy" (CNM) for the autosomal forms. Autosomal CNMs comprise a wide spectrum of phenotypes, ranging from severe neonatal to mild late-onset familial forms. The most frequent clinical features are delayed motor milestones, facial and generalised muscle weakness, ptosis and ophthalmoparesis or ophthalmoplegia [START_REF] Fardeau | Congenital myopathies[END_REF][START_REF] Jeannet | TClinical and histologic findings in autosomal centronuclear myopathy[END_REF] . The most prominent histopathological features consist of a high frequency of centrally located nuclei in the muscle fibres and predominance and hypotrophy of type 1 fibres [START_REF] Fardeau | Congenital myopathies[END_REF][START_REF] Jeannet | TClinical and histologic findings in autosomal centronuclear myopathy[END_REF][START_REF] Fischer | Characterization of the muscle involvement in dynamin 2 related centronuclear myopathy[END_REF] .

We identified the first mutations responsible for autosomal dominant (AD) CNM in the DNM2 gene encoding dynamin 2 (DNM2) [START_REF] Bitoun | Mutations in dynamin 2 cause dominant Centronuclear Myopathy[END_REF] , a large GTPase that was mainly implicated in endocytosis and membrane trafficking [START_REF] Jones | Role of dynamin in the formation of transport vesicles from the trans-Golgi network[END_REF][START_REF] Praefcke | The dynamin superfamily: universal membrane tubulation and fission molecules?[END_REF] . These first 4 heterozygous mutations were all restricted to the middle domain of the protein and were identified in families or sporadic cases mostly showing mild, late onset CNM [START_REF] Bitoun | Mutations in dynamin 2 cause dominant Centronuclear Myopathy[END_REF] . Independently, 3 heterozygous mutations in the Pleckstrin Homology (PH) domain of DNM2 have also been reported in dominant intermediate Charcot-Marie-Tooth disease type B (DI-CMTB) [START_REF] Züchner | Mutations in the pleckstrin homology domain of dynamin 2 cause dominant intermediate Charcot-Marie-Tooth disease[END_REF] .

Here, we report five patients with four novel heterozygous mutations in the PH domain of the DNM2 leading to sporadic CNM. In contrast to the previously reported DNM2 mutations, these four mutations are associated with a neonatal onset and a more severe phenotype.

Methods

DNM2 sequencing. The genomic sequence of DNM2 (NT_011295) was used to design intronic primers to amplify the 22 DNM2 exons, including exons 10bis and 13bis. Polymerase chain reactions and sequencing were done as previously described [START_REF] Bitoun | Mutations in dynamin 2 cause dominant Centronuclear Myopathy[END_REF] .

Clinical and morphological studies. The clinical data of the five patients were retrospectively reviewed and were summarized in the Table Clinical data. For the 5 patients, there was no family history of neuromuscular disorder. The clinical features are summarized in Table 1. The pregnancies were normal except for the mother of patient 5 affected by pregnancy-induced hypertension. The 5 patients presented with neonatal hypotonia. None needed respiratory support although three transiently received oxygen in neonatal period. They all had weak suckling and three required nasogastric tube for feeding. Three patients presented with marked facial weakness, open mouth and arched palate in early childhood and four have ptosis and ophthalmoparesis (Figure 1). Motor milestones were delayed in three patients. Patient 3 was not able to sit, nor control his head or lift his legs at the age of 1 year and patient 2 walked at the age of 28 months but only with aid. Only patients 4 and 5 reached independent ambulation at a normal age. All demonstrated generalized muscle weakness prominent in lower limbs, and distal limb muscles were more severely affected than proximal muscles. Serum CK level was slightly increased only in patient 3. Neutrophil counts available for patients 1, 2 and 3 were within the normal range for age. There were no cardiac abnormalities on electrocardiogram and echocardiography in patients 2 to 5, while mild apex dilatation was reported in patient 1. Patients 1, 2 and 3 have no respiratory involvement but patients 4 and 5 developed a restrictive respiratory syndrome at the age of 10 and 7 years, respectively. Patient 2 has been studied twice by electromyography (EMG) and nerve conduction velocities (NCV) at the ages of 4 months and 2 years. These showed myopathic changes, normal peroneal and median NCV and normal CMAP amplitudes. Patient 3 was similarly evaluated at 11 days after birth. EMG revealed a myopathic pattern and motor and sensory nerve conduction studies were normal (Table 1). Patient 5 also showed a myopathic pattern on EMG at the age of 5 years and had normal motor NCV.

Morphological data. Quadriceps or deltoid muscle biopsies performed from the ages 5 months to 8 years were available for the 5 patients. All biopsies showed predominance and hypotrophy of type 1 fibres with central nuclei (Figure 2). Percentage of fibres with central nuclei was 50%, 41%, 44%, and 3%, respectively for patients 1, 2, 3, and 5. NADH-TR reaction occasionally showed increased oxidative activity around central nuclei. A radial arrangement of sarcoplasmic strands around the central nuclei was observed in only a few fibers, mainly in patient 2. In the quadriceps biopsy from patient 3 performed at the age of 5 months, no radial arrangement of sarcoplasmic strands was observed, even by electron microscopy. Endomysial fibrosis was observed in the biopsies from patients 1 and 2 (Figure 2), but there was no evidence of necrosis or regeneration in any of the biopsies.

Discussion

In the present study, we identified four novel heterozygous mutations in the Pleckstrin Homology domain of the Dynamin 2 in young CNM patients who presented with a more severe phenotype than is typical in autosomal dominant CNM.

The four girls and the boy presently described, shared some XLMTM features [START_REF] Wallgren-Pettersson | The myotubular myopathies: differential diagnosis of the X linked recessive, autosomal dominant, and autosomal recessive forms and present state of DNA studies[END_REF][START_REF] Pierson | X-linked myotubular and centronuclear myopathies[END_REF] such as severe generalized hypotonia and muscle weakness at birth, ophthalmoparesis and facial weakness, and normal serum creatine kinase levels, nerve conduction velocities and cardiac function. However, in these DNM2-related CNM patients, there were no decreased foetal movements, premature births or long periods of respiratory insufficiency after birth, while these are often associated with XLMTM. Finally, DNM2-related neonatal CNM is associated with a relatively good prognosis in comparison with XLMTM in which most of patients die during the first yearsof life from respiratory failure. In our series, only the boy was still very weak at the age of one year while the girls progressively became stronger.

However, the two oldest patients developed a restrictive respiratory syndrome at the age of 7 and 10 years and lost deep tendon reflexes.

In the presently described patients, the classical morphological abnormalities described in late-onset CNM patients, especially concerning the radial arrangement of sarcoplasmic strands, were less pronounced. One can hypothesise that their scarcity is probably linked to the young age of the patients. However, in spite of the diagnosis difficulties, association of central nuclei and type 1 fibre predominance and hypotrophy with the described clinical findings allowed us to identify a group of 7 CNM-patients and we found DNM2 mutations in five of them. These results show a genetic heterogeneity in the autosomal neonatal CNM with a high frequency of mutation of the DNM2 gene and may enable a better selection for direct genetic testing.

Interestingly, three other DNM2 mutations in the PH domain have been reported in dominant intermediate Charcot-Marie-Tooth disease type B (DI-CMTB) [START_REF] Züchner | Mutations in the pleckstrin homology domain of dynamin 2 cause dominant intermediate Charcot-Marie-Tooth disease[END_REF] , which raises the possibility that these phenotypes may overlap. Previously, we found mild signs of axonal sensorimotor neuropathy in addition to predominant myopathic changes in 4 of 10 patients affected by mild, late-onset CNM harbouring mutations in the middle domain of DNM2 [START_REF] Fischer | Characterization of the muscle involvement in dynamin 2 related centronuclear myopathy[END_REF] . In the young patients reported here, the available electrophysiological data only show myopathic changes. It would be certainly of interest to follow the evolution of neurophysiological parameters in these patients in order to determine a possible progressive peripheral nerve involvement.

In conclusion, we report the first mutations in the PH domain of DNM2 in 5 patients with a neonatal onset and relatively severe CNM, with the development of a restrictive respiratory syndrome at the end of the first decade. These heterozygous de novo mutations enlarge the spectrum of DNM2-related CNM from the classical mild form to include the neonatal phenotype, overlapping with the mild end of the spectrum of XLMTM but with a better prognosis. Our findings show that DNM2 gene should be considered in severe sporadic neonatal CNM without MTM1 mutations in whom inheritance is impossible to predict.

Legends Table 1: Summary of clinical data

Results

  DNM2 screening. We identified 4 heterozygous mutations in 5 CNM patients by sequencing of the coding sequence of DNM2 gene. The mutations were all located in exon 16: a missense mutation (c.1852G>A) changing alanine 618 to threonine (p.A618T) in patient 1, a missense mutation (c.1856C>T) changing serine 619 to leucine (p.S619L) in patients 2 and 3, a missense mutation (c.1856C>G) changing serine 619 to tryptophan (p.S619W) in patient 4, and a three base pairs deletion (c.1873_1875delGTC) inducing deletion of the valine 625 (p.V625del) in patient 5. Sequences of DNA samples from parents, available for patients 1, 2,3 and 5, were normal and the paternity was confirmed, indicating de novo mutations. The four mutations were not found in 100 unrelated healthy control subjects. All mutations were detected in 3 conserved amino-acids located in the PH domain of DNM2 (Figure1A).

F

  : Female; M: Male, yrs: years, mo: months. -: no information. NVC: nerve conduction velocities, CMAP: compound muscle action potential, SNAP: sensory nerve action potential.

Figure 1 : 1 (

 11 Figure 1: Clinical features of patients 1, 2 and 3 with early onset CNM with DNM2

Figure 2 :

 2 Figure 2: Histological features in DNM2-related CNM patient

  . Motor and sensory nerve conduction studies

	were performed using standard procedure. Standard histochemical techniques were performed
	on deltoid or quadriceps muscle biopsies from the five patients (Hematein-Eosin, Myosin
	Adenosine Triphosphatase preincubated at pH 9.4 or 4.2, and Nicotinamide Adenine
	Dinucleotide-Tetrazolium Reductase staining). Digital photographs were obtained with a
	Zeiss AxioCam HRc attached to a Zeiss Axioplan Bright Field Microscope. For patients 1, 2
	and 3, muscle samples were also fixed in 2.5% glutaraldehyde and embedded in Polybed 812.
	Ultrathin sections were stained with uranyl acetate and lead citrate (Reynolds) and examined
	under a Zeiss 109 electron microscope.

We thank the patients and their families for their participation in this study, Dr M Lucero for assistance, Dr T Stojkovic and Dr K Flaningan for helpful advices, the Myopathies (AFM), and the Programme of Collaboration ECOS-SECyT (N° A02S02) for financial support. Jorge A. Bevilacqua was supported by the Programme Alban, the European Union Programme of High Level Scholarships for Latin America, scholarship No.E04E028343CL. Marc Bitoun was a recipient of an INSERM fellowship.