P Abrial

Y Moudden

J L Starck

B Afeyan

J Bobin

J Fadili

M K Nguyen

Morphological Component Analysis and Inpainting on the Sphere: Application in Physics and Astrophysics

Keywords: Spherical Data, Morphological Component Analysis, Sparse representation, Inpainting

Morphological Component Analysis (MCA) is a new method which takes advantage of the sparse representation of structured data in large overcomplete dictionaries to separate features in the data based on the diversity of their morphology. It is an efficient technique in such problems as separating an image into texture and piecewise smooth parts or for inpainting applications. The MCA algorithm consists of an iterative alternating projection and thresholding scheme, using a successively decreasing threshold towards zero with each iteration. In this paper, the MCA algorithm is extended to the analysis of spherical data maps as may occur in a number of areas such as geophysics, astrophysics or medical imaging. Practically, this extension is made possible thanks to the variety of recently developed transforms on the sphere including several multiscale transforms such as the undecimated isotropic wavelet transform on the sphere, the ridgelet and curvelet transforms on the sphere. An MCA-inpainting method is then directly extended to the case of spherical maps allowing us to treat problems where parts of the data are missing or corrupted. We demonstrate the usefulness of these new tools of spherical data analysis by focusing on a selection of challenging applications in physics and astrophysics.

Introduction

A usual task in processing signals, images as well as spherical data maps, is to decompose the data into its elementary building blocks. This can be formulated as an inverse problem where the data is assumed to have been generated according to the following model :

y = i α i φ i + η (1)
that is a linear combination of relevant waveforms φ i ∈ R n with weights α i . Here η represents possible contamination by additive, typically Gaussian white noise. Given data y ∈ R n , one then wants to recover the underlying structures that is to say estimate a set of waveforms φ i that build the data and their corresponding weights αi . The solution to this estimation problem will depend heavily on the available prior information. Of interest here is the case where one is given a priori a set a waveforms from which to select a good subset. This set may be a basis, a frame or several bases or frames grouped into a large redundant dictionary. Possible dictionaries in 1D and 2D include Fourier and related bases, wavelet bases, as well as other more recent multiscale systems such as the ridgelet [START_REF] Candès | Ridgelets: the key to high dimensional intermittency?[END_REF] and curvelet frames [START_REF] Donoho | Digital curvelet transform: strategy, implementation and experiments[END_REF][START_REF] Starck | The curvelet transform for image denoising[END_REF], etc. Depending on the morphology of the data, each of these dictionaries will have different performance characteristics in a non-linear approximation scheme. For instance, sparse approximations of piecewise smooth signals or images with point singularities are easily obtained using wavelets. However these are no longer optimal in the case of piecewise smooth images with singularities along smooth curves or edges. Such images are more efficiently approximated using curvelets which are highly anisotropic and thus exhibit high directional selectivity. Digital implementations of both ridgelet and curvelet transforms and their application to image denoising are described in [START_REF] Starck | The curvelet transform for image denoising[END_REF].

Available transforms in the spherical topology include the spherical harmonics and several wavelet transforms. Software packages such as Healpix1 [START_REF] Górski | HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere[END_REF] or Glesp [START_REF] Doroshkevich | Gauss-legendre sky pixelization (glesp) scheme for cmb maps[END_REF] provide approximate digital spherical harmonic transform routines based on their specific pixelization schemes. Schröder and Sweldens [START_REF] Schröder | Spherical wavelets: Efficiently representing functions on the sphere[END_REF] have developed an orthogonal wavelet transform on the sphere based on the Haar wavelet function which then suffers from the poor frequency domain localization properties of the primitive Haar function and from the problems inherent in orthogonal decomposition (e.g. lack of translation invariance). A few papers describe continuous wavelet transforms on the sphere [START_REF] Antoine | Wavelets on the 2-sphere: a group-theoretical approach[END_REF][START_REF] Cayón | Spherical Mexican hat wavelet: an application to detect non-Gaussianity in the COBE-DMR maps[END_REF][START_REF] Holschneider | Wavelet analysis on the sphere[END_REF][START_REF] Wiaux | Correspondence Principle between Spherical and Euclidean Wavelets[END_REF][START_REF] Bogdanova | Stereographic wavelet frames on the sphere[END_REF] which have been extended to directional wavelet transforms [START_REF] Wiaux | Fast directional correlation on the sphere with steerable filters[END_REF][START_REF] Mcewen | A fast directional continuous spherical wavelet transform[END_REF]. Although useful for data analysis, these continuous transforms lack an inverse transform and hence are clearly not suitable for restoration or synthesis purposes.

In their pioneering work, Freeden and Maier [START_REF] Freeden | On multiscale denoising of spherical functions: Basic theory and numerical aspects[END_REF][START_REF] Freeden | A survey on wavelet methods for (geo)applications[END_REF] gave a wavelet transform/reconstruction scheme on the sphere which is based on the spherical harmonic transform. Following this idea, Starck et al. [START_REF] Starck | Wavelets, ridgelets and curvelets on the sphere[END_REF] have proposed a new invertible isotropic undecimated wavelet transform (UWT) on the sphere which preserves the same desirable properties as the standard isotropic UWT for flat 2D maps [START_REF] Starck | Image Processing and Data Analysis: The Multiscale Approach[END_REF]: the reconstruction is simple and immediate since it is just the addition of all the wavelet bands with the coarsest scale. Based on this new decomposition, other multiscale transforms such as the pyramidal wavelet transform, the ridgelet transform and the curvelet transform have been successfully constructed on the sphere [START_REF] Starck | Wavelets, ridgelets and curvelets on the sphere[END_REF]. Each of these decompositions on the sphere will sparsely represent parts of the image based on their morphological properties. Wavelets will easily detect more or less isotropic localized structures, while curvelets are better suited for efficiently detecting highly anisotropic objects.

A data set y has an exact representation over any complete basis of the data space, or several such exact representations in the case of redundant overcomplete dictionaries. However, these representations are not equally interesting in terms of data modeling or feature detection. In fact, a strong a priori is to favor representations of y that use only a small number of waveforms leading to a more concise and possibly more interpretable representation of the data. In fact, building sparse representations or approximations is the (he)art of structured data processing: the design of good detection, denoising, restoration and compression algorithms relies on the availability of good dictionaries and good selection algorithms. Indeed, selecting the smallest subset of waveforms from a large dictionary, that will linearly combine to reproduce the salient features of a given signal or image, is a hard combinatorial problem. Several pursuit algorithms have been proposed that can help build very sparse decompositions such as the greedy Matching Pursuit (MP) [START_REF] Mallat | Matching pursuit with time-frequency dictionaries[END_REF] algorithm which refines the signal approximation by picking at each iteration the one waveform which best correlates with the current approximation error. Basis Pursuit (BP) [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF] is a global procedure which seeks an approximation ỹ to y by solving the linear programming problem:

min α α ℓ 1 subject to y = Φα. (2)
where the ℓ 1 norm measures sparsity in place of the ℓ 0 counting norm. In the presence of noise, a noise-aware variant of BP, known as BPDN (for BP denoising), can be stated as a convex quadratic programming problem and solved using the Interior Point method [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF]. The BPDN problem can also be written in the augmented Lagrangian form:

min α y -Φα 2 ℓ 2 + λ • α ℓ 1 (3)
Among all possible solutions, the chosen one has the minimum ℓ 1 norm. This choice of ℓ 1 norm is very important. An ℓ 2 norm, as used in the method of frames [START_REF] Daubechies | Time-frequency localization operators: A geometric phase space approach[END_REF], does not favor sparsity [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF]. A number of recent results prove that these algorithms will recover the unique maximally sparse decomposition provided this solution is sparse enough and the dictionnary is sufficiently incoherent [START_REF] Donoho | Optimally sparse representation in general (non-orthogonal) dictionaries via ℓ 1 minimization[END_REF][START_REF] Elad | A generalized uncertainty principle and sparse representation in pairs of bases[END_REF][START_REF] Gribonval | Sparse representations in unions of bases[END_REF][START_REF] Donoho | Stable recovery of sparse overcomplete representations in the presence of noise[END_REF][START_REF] Fuchs | Recovery of exact sparse representations in the presence of bounded noise[END_REF]. Nevertheless, in problems involving large data sets (e.g. images, spherical maps), BP or MP synthesis algorithms are computationally prohibitive. Morphological Component Analysis (MCA) is a recent faster alternative described in [START_REF] Starck | Redundant multiscale transforms and their application for morphological component analysis[END_REF] that constructs a sparse representation of a signal or an image assuming that it is a combination of morphologically distinct features which are sparsely represented in different dictionaries associated with fast transform algorithms. For instance, images commonly combine contours and textures: the former are well accounted for using curvelets, while the latter may be well represented using local cosine functions. In searching for a sparse decomposition of a signal or image y, it is assumed that y is a sum of K components (s k ) 1,...,K , where each can be described as s k = Φ k α k with a possibly over-complete dictionary Φ k and a sparse vector of coefficients α k . It is further assumed that for any given component the sparsest decomposition over the proper dictionary yields a highly sparse description, while its decomposition over the other dictionaries, Φ k ′ =k , is non sparse. Thus, the different Φ k can be seen as discriminating between the different components of the initial signal. MCA achieves its sparse decomposition relying on an iterative thresholding algorithm with a successively decreasing threshold [START_REF] Bobin | Morphological component analysis: An adaptive thresholding strategy[END_REF] thus refining the current approximation by including finer structures alternatingly in the different morphological components. Based on MCA, it has also been shown that we can derive a very efficient inpainting method [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF].

This paper: Motivated by the success of MCA in signal and image processing, the purpose of this contribution is to take advantage of the variety of transforms on the sphere recently made available [START_REF] Starck | Wavelets, ridgelets and curvelets on the sphere[END_REF] to extend the applicability of MCA to the analysis of spherical maps which are commonly recorded in a number of areas such as geophysics, astrophysics or medical imaging. As in the case of Euclidean 2D images, we further extend the MCA algorithm on the sphere in order to perform inpainting tasks on the sphere. The proposed numerical tools are shown to be valuable in several selected applications in physics and astrophysics. The construction of the undecimated isotropic wavelet and curvelet transforms on the sphere is reviewed in the next section. Sections 3 and 4 describe the extension to the sphere of the MCA algorithm and of its modification for inpainting purposes.

Transforms on the Sphere

Isotropic Undecimated Wavelet Transform on the Sphere (UWTS)

The undecimated isotropic transform on the sphere described in [START_REF] Starck | Wavelets, ridgelets and curvelets on the sphere[END_REF] is similar in many respects to the regular à trous isotropic wavelet transform. It is obtained using a zonal scaling function φ lc (ϑ, ϕ) which depends only on colatitude ϑ and is invariant with respect to a change in longitude ϕ. It follows that the spherical harmonic coefficients φlc (l, m) of φ lc vanish when m = 0 which makes it simple to compute the spherical harmonic coefficients ĉ0 (l, m) of c 0 = φ lc * f where * stands for convolution :

ĉ0 (l, m) = φ lc * f (l, m) = 4π 2l + 1 φlc (l, 0) f (l, m) (4)
A possible scaling function [START_REF] Starck | Image Processing and Data Analysis: The Multiscale Approach[END_REF], defined in the spherical harmonics representation, is

φ lc (l, m) = 2 3 B 3 ( 2l lc )
where B 3 is the cubic B-spline compactly supported over [-2, 2]. Denoting φ 2 -j lc a rescaled version of φ lc with cut-off frequency 2 -j l c , a multi-resolution decomposition of f on a dyadic scale is obtained recursively :

c 0 = φ lc * f c j = φ 2 -j lc * f = c j-1 * h j-1 (5)
where the zonal low pass filters h j are defined by

Ĥj (l, m) = 4π 2l + 1 ĥj (l, m) =      φ lc 2 j+1 (l,m) φ lc 2 j (l,m) if l < lc 2 j+1
and m = 0 0 otherwise

The cut-off frequency is reduced by a factor of 2 at each step so that in applications where this is useful such as compression, the number of samples could be reduced by the same factor. Using a pixelization scheme such as Healpix [START_REF] Górski | HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere[END_REF], this can easily be done by dividing by 2 the Healpix nside parameter when computing the inverse spherical harmonics transform. Of course, this is only an approximate Sampling Theorem but it proved sufficient for numerical purposes. However, in the present isotropic undecimated transform, no downsampling is performed and the maps have the same number of pixels on each scale. Hence the orthogonality requirement is relaxed, which provides us with a higher degree of freedom in the choice and design of the wavelet function ψ lc to be used with the scaling function φ lc . As in the à trous algorithm, the wavelet coefficients can be defined as the difference between two consecutive resolutions, w j+1 (ϑ, ϕ) = c j (ϑ, ϕ) -c j+1 (ϑ, ϕ) which defines a zonal wavelet function ψ lc as

ψ lc 2 j (l, m) = φ lc 2 j-1 (l, m) -φ lc 2 j (l, m) (7)
This particular decomposition is readily inverted by summing the coefficient maps on all wavelet scales

f (ϑ, ϕ) = c J (ϑ, ϕ) + J j=1 w j (ϑ, ϕ) (8)
where we have made the simplifying assumption that f is equal to c 0 . Obviously, other wavelet functions could be used just as well. Also, because of the redundancy of the described decomposition, the inverse transform is not unique and in fact this can profitably be used to impose additional constraints on the synthesis functions (e.g. smoothness, positivity) used in the reconstruction [START_REF] Starck | The undecimated wavelet decomposition and its reconstruction[END_REF].

Curvelets on the Sphere

The 2D ridgelet transform [START_REF] Candès | Ridgelets: the key to high dimensional intermittency?[END_REF] was developed in an attempt to overcome some limitation inherent in former multiscale methods e.g. the 2D wavelet, when handling smooth images with edges i.e. singularities along smooth curves. Ridgelets are translation invariant ridge functions with a wavelet profile in the normal direction. Although ridgelets provide sparse representations of smooth images with straight edges, they fail to efficiently handle edges along curved lines. This is the framework for curvelets which were given a first mathematical description in [START_REF] Candès | Curvelets -a surprisingly effective nonadaptive representation for objects with edges[END_REF]. Basically, the curvelet dictionary is a multiscale pyramid of localized directional functions with anisotropic support obeying a specific parabolic scaling such that at scale 2 -j , its length is 2 -j/2 and its width is 2 -j . This is motivated by the parabolic scaling property of smooth curves. Other properties of the curvelet transform and decisive optimality results in approximation theory are reported in [START_REF] Candès | Curvelets -a surprisingly effective nonadaptive representation for objects with edges[END_REF][START_REF] Candès | Curvelets and curvilinear integrals[END_REF]. Notably, curvelets provide optimally sparse representations of manifolds which are smooth away from edge singularities along smooth curves. Several digital curvelet transforms [START_REF] Donoho | Digital curvelet transform: strategy, implementation and experiments[END_REF][START_REF] Starck | The curvelet transform for image denoising[END_REF][START_REF] Do | The contourlet transform: an efficient directional multiresolution image representation[END_REF][START_REF] Candes | Fast discrete curvelet transforms[END_REF] have been proposed which attempt to preserve the essential properties of the continuous curvelet transform and many papers [START_REF] Starck | Detecting cosmological non-gaussian signatures by multiscale methods[END_REF][START_REF] Starck | Redundant multiscale transforms and their application for morphological component analysis[END_REF][START_REF] Herrmann | The contourlet transform: an efficient directional multiresolution image representation[END_REF][START_REF] Starck | Gray and color image contrast enhancement by the curvelet transform[END_REF][START_REF] Starck | Astronomical image representation by the curvelet tansform[END_REF] report on their successful application in image processing experiments. The contourlet transform in [START_REF] Do | The contourlet transform: an efficient directional multiresolution image representation[END_REF] is a filter bank implementation of the curvelet transform. Candès et al. implement fast discrete curvelet transforms in [START_REF] Candes | Fast discrete curvelet transforms[END_REF]. Their design is very close to the mathematical transformation as they resort to a paving of the Fourier plane into pseudo-polar wedges delimited by dyadic concentric squares. The so-called first generation discrete curvelet described in [START_REF] Donoho | Digital curvelet transform: strategy, implementation and experiments[END_REF][START_REF] Starck | The curvelet transform for image denoising[END_REF] is a very different construction : it consists in applying the ridgelet transform to sub-images of a wavelet decomposition of the original image. By construction, the sub-images are well localized in space and frequency and the subsequent ridgelet transform provides the necessary directional sensitivity. This latter implementation in combination with the good geometric properties of the Healpix pixelisation scheme, inspired the digital curvelet transform on the sphere [START_REF] Starck | Wavelets, ridgelets and curvelets on the sphere[END_REF].

Partitioning using the Healpix representation. The Healpix representation [START_REF] Górski | HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere[END_REF] is a curvilinear partition of the sphere into quadrilateral pixels of exactly equal area but with varying shape. The base resolution divides the sphere into 12 quadrilateral faces of equal area placed on three rings around the poles and equator. Each face is subsequently divided into nside 2 pixels following a hierarchical quadrilateral tree structure. The geometry of the Healpix sampling grid makes it easy to partition a spherical map into blocks of a specified size. We first extract the twelve base-resolution faces, and each face is then decomposed into smoothly (i.e. an apodizing window is used) overlapping blocks of side-length B j pixels in such a way that the overlap between two vertically adjacent blocks is a rectangular array of size B j × B j /2, as in the 2D digital curvelet transform [START_REF] Donoho | Digital curvelet transform: strategy, implementation and experiments[END_REF][START_REF] Starck | The curvelet transform for image denoising[END_REF][START_REF] Starck | Astronomical image representation by the curvelet tansform[END_REF]. With this scheme however, there is no overlapping between blocks belonging to different base-resolution faces. This may result in blocking aritfacts in different data processing tasks such as image denoising via non linear filtering. A simple way around this difficulty, which however results in greater redundancy, is to work with various rotations of the data with respect to the sampling grid.

Ridgelet transform

The continuous ridgelet transform is the application of a 1D-wavelet transform to the angular slices of the Radon transform [START_REF] Candès | Ridgelets: the key to high dimensional intermittency?[END_REF]. A digital implementation described in [START_REF] Starck | Astronomical image representation by the curvelet tansform[END_REF] consists in i) first computing the 2D Fourier transform of tthe image, ii) next extracting lines going through the origin in the frequency plane, iii) then computing the 1D inverse Fourier transform of each of these lines, iv) and finally computing the 1D wavelet transform of the extracted lines. The first three steps implement the discrete Radon transform method called the linogram. Other implementations of the Radon transform, such as the Slant Stack Radon Transform [START_REF] Donoho | Digital Ridgelet Transform Based on True Ridge Functions[END_REF], can be used as well, provided they offer an exact reconstruction. Applying the ridgelet transform on sub-images as suggested in [START_REF] Candès | Harmonic analysis of neural netwoks[END_REF] provides the means to localize and limit the extent of the ridgelet functions. More details on the implementation of the digital ridgelet transform can be found in [START_REF] Starck | The curvelet transform for image denoising[END_REF].

Curvelet transform Finally, the proposed digital curvelet transform algorithm on the sphere is as follows:

1. Apply the undecimated isotropic wavelet transform on the sphere with J scales, 2. Initialize the block size B1 = Bmin, 3. For j = 1, . . . , J do,

• partition the subband wj with a block size Bj and apply the digital ridgelet transform to each block,

• if j modulo 2 = 1 then Bj+1 = 2Bj , • else Bj+1 = Bj .
where smaller j means smaller length scales and higher frequency bands. Doubling the sidelength of the localizing window at every other dyadic sub-band (i.e. when the spatial frequency has been divided by four ) approximately preserves the specific parabolic scaling property of the continuous curvelets. Nevertheless, the proportionality factor between width and length 2 has to be decided on in practice. The proposed implementation uses the default value B min = 16 pixels. Figure 1 shows the backprojection of a few curvelet coefficients at different scales and orientations on the sphere. The digital curvelet transform on the sphere is clearly invertible in the sense that each step of the overall transform is itself invertible. The curvelet transform on the sphere has a redundancy factor of 16J + 1 when J scales are used, which may be a problem for handling huge data sets such as from the future Planck-Surveyor experiment. This can be reduced by substituting the pyramidal wavelet transform to the undecimated wavelet transform in the above algorithm. More details on the wavelet, ridgelet, curvelet algorithms on the sphere can be found in [START_REF] Starck | Wavelets, ridgelets and curvelets on the sphere[END_REF] and software related to these new transforms is available from the web page http://jstarck.free.fr.

Local DCT on the Sphere The Healpix partitionning of the sphere into warped square blocks allows for even more numerical local transformations on the sphere. For instance, a local DCT on the sphere is readily obtained by applying the standard DCT to a complete set of possibly overlapping blocks of maybe different sizes. This heuristic revealed useful in a particular application described in section 3.3. Although it is difficult to fully characterize the geometry of such digital transformations on the sphere, they profitably enrich the set of tools available for spherical data analysis.

3 MCA on the Sphere

Principle and algorithm

For a given spherical map y modeled as a linear combination of K spherical maps

(s k ) k=1,••• ,K , y = K k=1 s k , having different morphologies, MCA assumes that a dictionary of bases {Φ 1 , • • • , Φ K } exists such that, for each k, s k is sparse in Φ k while its representation in the other Φ k ′ ( k ′ = k) is not sparse : ∀k ′ = k, ||Φ T k s k || 0 < ||Φ T k ′ s k || 0
, where ||x|| 0 denotes the ℓ 0 pseudo-norm of the vector x . The problem

While sensible from the point of view of the desired solution, the problem formulated in Equation ( 9) is non-convex and combinatorial by nature. Its complexity grows exponentially with the number of columns in the overall dictionary (NP-hard problem). Motivated by recent equivalence results e.g. in [START_REF] Donoho | Optimally sparse representation in general (non-orthogonal) dictionaries via ℓ 1 minimization[END_REF], the MCA algorithm seeks a solution to the following minimization problem:

min s 1 ,...,s K λ K k=1 α k 1 + y - K k=1 s k 2 2 with s k = Φ k α k ( 10
)
where an ℓ 1 sparsity measure is substituted to the ℓ 0 counting norm following a prescription of the Basis Pursuit algorithm [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF]. In the above, the equality constraint was relaxed and again s k = Φ k α k . In the case where each Φ k is an orthonormal basis, a block-coordinate solution to the above problem is given by the following set of coupled equations:

∀k, s k = r k - λ k 2 Φ k sign(Φ T k s k ) with r k = s - k ′ =k s k ′ (11)
This can be solved efficiently using the iterative Block-Coordinate Relaxation Method [START_REF] Bruce | Block coordinate relaxation methods for nonparametric signal de-noising[END_REF] in conjunction with, at a given k, a soft-thresholding of the decomposition of r k over Φ k . However, when non-unitary or redundant transforms are used, the above is no longer strictly valid. Nevertheless, simple shrinkage still gives satisfactory results as explained in [START_REF] Elad | Why simple shrinkage is still relevant for redundant representations[END_REF]. Finally, denoting by T k and R k the forward and inverse transforms associated with the redundant dictionary Φ k , MCA seeks a solution to problem [START_REF] Bouchet | Foregrounds and CMB experiments I. Semi-analytical estimates of contamination[END_REF] with the following algorithm:

1. Set the number of iterations Imax and the initial thresholds " λ

(0) k " k 2. While λ (t)
k is greater than a given lower bound λmin (e.g. can depend on the noise standard deviation), -Proceed with the following iteration to estimate components (s k ) k=1,...,K at iteration t:

For k = 1, • • • , K • Compute the residual term r (t)
k assuming the current estimates

s(t-1) k ′ =k of s k ′ =k , are fixed: r (t) k = y - P k ′ =k s(t-1)
k ′

• Estimate the current coefficients of s(t) k by thresholding with threshold λ

(t) k : α(t) k = δ λ (t) k " T k r (t) k "
• Get the new estimate of s k by reconstructing from the selected coeffcients

α(t) k : s(t) k = R k α(t)
k -Decrease the thresholds λ k following a given strategy.

Thresholding strategy

The operator δ in the above algorithm is a soft thresholding operator as a result of the use of an ℓ 1 sparsity measure in approximation to the ideal ℓ 0 norm. In practice, hard thresholding leads generally to better results [START_REF] Starck | Redundant multiscale transforms and their application for morphological component analysis[END_REF]. The final threshold should vanish in the noiseless case or it may be set to a multiple of the noise standard deviation in the presence of noise as in common detection or denoising methods. The way the threshold is decreased along the iterations of the proposed iterative thresholding scheme is paramount in terms of performance of the MCA separation mechanism. The original algorithm [START_REF] Starck | Redundant multiscale transforms and their application for morphological component analysis[END_REF] used a linear strategy :

λ (t) = λ (0) -(t -1) λ (0) -λ min I max -1 (12)
where λ (0) is the initial threshold, and I max is the number of iterations. The first threshold can be set automatically to a large enough value such as the maximum of all coefficients λ (0) = max k T k y ∞ . But there is no way to estimate the minimum number of iterations yielding a successful separation. Too small a number of iterations leads to bad separation while too large a number is computationally costly. Further, experiments have clearly shown that the optimal number of iterations depends on the data.

We recently focused on devising some new data adaptive thresholding strategies to speed up the MCA decomposition preserving the quality of the component separation. Hereafter we describe two promising strategies, namely MAD and MOM, in the case where K = 2 ; generalizing to K ≥ 2 is straightforward.

MAD Consider a map y such that y = s 1 + s 2 = Φ 1 α 1 + Φ 2 α 2 where s 1 and s 2 have similar ℓ 2 norm and α k=1,2 = Φ T k=1,2 s k=1,2 are sparse. When both Φ k=1,2 are orthonormal bases, decomposing y in Φ 1 leads to yΦ T

1 = α 1 + Φ T 1 Φ 2 α 2 .
Provided the mutual coherence [START_REF] Bruckstein | A generalized uncertainty principle and sparse representation in pairs of r n bases[END_REF][START_REF] Gribonval | Sparse representations in unions of bases[END_REF][START_REF] Donoho | Optimally sparse representation in general (non-orthogonal) dictionaries via ℓ 1 minimization[END_REF] of Φ 1 and Φ 2 is low, y 2 has no particular structure in Φ 1 and hence it is tempting to model Φ T 1 s 2 as a Gaussian noise. Its standard deviation can be estimated using a robust estimator such as the Median Absolute Deviation (MAD) [START_REF] Donoho | Ideal spatial adaptation via wavelet shrinkage[END_REF]. It follows that estimating the significant entries α1 in α 1 is a denoising problem readily solved by thresholding Φ T 1 y with a threshold kσ (typically k is in the range 3 to 4). The next step is to project the residual y -s1 = y -Φ 1 α1 on Φ 2 and so on. Clearly, the variance of the residual decreases along iterations and so this provides a simple strategy to adaptively control the threshold in the MCA algorithm. In practice, this strategy remains fruitful in the case of redundant dictionaries. Donoho et al. in [START_REF] Donoho | Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit[END_REF] have recently focused on an iterative thresholding scheme applied to solving under-determined linear sparse problems in which they use a similar rule to manage their decreasing threshold.

MOM Let s(t)

1 and s(t) 2 denote the current estimates of components s 1 and s 2 at the t th iteration of the MCA decomposition of y. The current residual is r

(t) = y - s(t) 1 - s(t) 2 .
In the strategy coined MOM as in "Mean of Max", the value of the threshold at iteration t is given by :

λ (t) = 1 2 ||Φ T 1 y - s(t-1) 1 - s(t-1) 2 || ∞ + ||Φ T 2 y - s(t-1) 1 - s(t-1) 2 || ∞ (13)
which is easily computed at each step of the iterative process. When one considers more than two dictionaries, one should take the mean of the two largest decomposition coefficients of the full residual over two distinct dictionaries. The intuition underlying this strategy is that the next significant coefficients to be selected should be attached to the dictionary in which the projection of the full residual has coefficients of largest amplitudes. Assuming the coefficients selected at iteration t are in Φ 1 , it can be shown, under some conditions on the sparsity of the components and the mutual coherence of the dictionary [START_REF] Bobin | Morphological component analysis: An adaptive thresholding strategy[END_REF], that the proposed strategy fixes the threshold so that :

||Φ T 1 Φ 2 ᾱ(t-1) 2 || ∞ < λ (t) 1 < ||ᾱ (t-1) 1 || ∞ , ᾱ(t-1) k=1,2 = α k=1,2 - α(t-1) k=1,2 (14)
hence avoiding false detections (upper bound) and ensuring that at least one coefficient is selected (lower bound). This thresholding strategy can easily be made more or less conservative depending on the desired decomposition speed. With these new thresholding strategies, MCA is a fast and robust algorithm to achieve sparse decompositions in redundant dictionaries and a practical alternative to other well-known sparse decomposition algorithms [START_REF] Bobin | Morphological component analysis: An adaptive thresholding strategy[END_REF].

Example

The spherical maps shown on figure 2 illustrate a simple numerical experiment. We applied the proposed Morphological Component Analysis on the Sphere to synthetic data resulting from the linear mixture of components respectively sparse in the spherical harmonics and the isotropic wavelet representations. The method was able to separate the data back into its original constituents. A more involved application is described in the next section.

Application in Physics

In Inertial Confinement Fusion (ICF) a spherical shell is irradiated by laser energy directly or after the laser energy has been converted to soft X-rays [START_REF] Atzeni | The Physics of Inertial Fusion[END_REF]. Either way, the aim is to implode the capsule which contains a shell of nuclear fusion fuel (deuterium and tritium) ready to ignite if, after it has been imploded, its density is high enough and a hot spot in its center becomes hot enough to cause a propagating nuclear burn wave to travel through the rest of the fuel. This ultimate energy source will not work if during the implosion hydrodynamic instabilities develop which can break apart the shell before it assembles at the center and a hot spot forms [START_REF] Lindl | Inertial Confinement Fusion: The Quest for Ignition and Energy Gain Using Indirect Drive[END_REF]. Hydrodynamic instabilities such as Rayleigh-Taylor occur due to nonuniformities in the laser spatial profile or imperfections in the composition of multiple surfaces which make up the layers of thin material that surround the nuclear fuel. Very small amplitude imperfections initially can result in the ultimate failure of the target due to the large compression ratios involved in ICF. It is therefore extremely important to characterize the inner and outer surfaces of ICF shell targets so as to know whether they are worthy of consideration for ICF implosions. One day in a reactor setting tens of thousands of targets will have to be imploded daily so that checking each one is totally out of the question. Instead, very good target fabrication quality control processes have to be adopted so that confidence levels in proper performance will be high. A major step along this path to fusion energy then is to understand why imperfections occur and to correct the systematic elements and control the harm done by random sources. Fine structures on the surfaces of spherical shells can be measured on the nanometer scale, among others, by atomic force microscopy or phase shifting spherical diffractive optical interferometry. An example of such measurements is shown on figure 3. As can be seen from the figure, there appears to be a superposition of global scale variations, isolated bumps and scratches as well as artifacts which look like interference patterns on intermediate scales of localization. The latter must be isolated and eliminated from consideration when deciding the readiness of the target for implosion. We have achieved the morphological feature separation by first doing an isotropic wavelet transform on the spherical data and subtracting the coarsest scale information. MCA on the sphere was used on the rest of the image using the undecimated wavelet and the local cosine transforms on the sphere. The isolated bumps were thus identified and the measurement technique caused artifacts were removed easily. The resulting bumps added to the coarsest scale, is the clean data with the interference patterns and artifacts removed as shown in figure 4. The spherical harmonic decomposition of the cleaned image gives rise to coefficients of various ℓ modes which will be amplified by the implosion process which can now be assessed correctly using numerical hydrodynamics simulation generated growth factors. If the bumps are clustered and not randomly distributed, then systematic errors in the manufacturing process can be tracked down. A code called MODEM has been put together to study such target surface data and extract the localized bump statistics including their correlations in height, size and relative location. For more details see [START_REF] Afeyan | Modem: Morphological diversity extraction method to identify, classify and characterize surface defects of spherical icf targets via afm and phase shifting diffraction interferometry[END_REF].

Inpainting on the Sphere

Algorithm

Named after the expert recovery process used for the restoration of deteriorated masterpieces, inpainting refers to a set of techniques used to alter images in a way that is undetectable to people who are unaware of the original images. There are numerous applications among which removing scratches or objects in digitized photographs, removing overlayed text or graphics, filling-in missing blocks in unreliably transmitted images, predicting values in images for better compression or image upsampling. Inpainting algorithms strive to interpolate through the gaps in the image relying on the available pixels, the continuation of edges, the periodicity of textures, etc. The preservation of edges and texture, in other words discontinuities, across gaps has attracted much interest, and many contributions have been proposed to solve this interpolation task. Non-texture image inpainting has received considerable interest and excitement since the pioneering paper by Masnou and Morel [START_REF] Masnou | Level lines based disocclusion[END_REF][START_REF] Masnou | Disocclusion: A variational approach using level lines[END_REF] who proposed variational principles for image disocclusion. A recent wave of interest in inpainting has started from the recent contributions of Sapiro et al. [START_REF] Ballester | Filling-in by joint interpolation of vector fields and grey levels[END_REF][START_REF] Bertalmio | Navier-stokes,fluid dynamics, and image and video inpainting[END_REF][START_REF] Bertalmio | Image inpainting[END_REF], followed by Chan and Shen [START_REF] Chan | Local inpainting models and tv inpainting[END_REF]. In these works, authors point to the importance of geometry and design anisotropic diffusion PDEs to fill in gaps by smooth continuation of isophotes. PDE methods have been shown to perform well on piecewise smooth functions.

A very different approach is the inpainting algorithm based on MCA described in [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF] which has proved capable of filling in holes in either texture or cartoon content in 2D images. To make the link Figure 4: top : Spherical map obtained by subtracting the coarse scale map on the right of figure 3 from the initial map on the left of figure 3. bottom : Component maps separated by the MCA method on the sphere : interference patterns and measurement artifacts were grabbed by the local cosine functions on the sphere (left) while the isolated bumps were caught using the undecimated wavelet on the sphere (right). Adding back the coarse scale on the right of figure 3 to the latter map results in a clean map of the surface structures of an ICF spherical shell with the interference patterns and artifacts removed. between building sparse representations and inpainting, consider the effect of a rectangular gap on the set of Fourier coefficients of a monochromatic sinewave : because of the non-locality of the Fourier basis functions it takes a large number of coefficients to account for the gap, which is known as the Gibbs effect. Seeking a sparse representation of the incomplete sine-wave outside the gap, that is without fitting the gap, enables the recovery of the complete monochromatic sinewave.

Following [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF], an inpainting algorithm on the sphere is readily built from the Morphological Component Analysis on the sphere described in the previous section. Consider a discrete spherical data map y and a binary map M such that ones in M indicate that the corresponding pixels in y are valid data while zeros indicate invalid data. The objective function of MCA (eq. 10) can be modified as follows :

min s 1 ,...,sn λ K k=1 α k 1 + M ⊙ (y - K k=1 s k ) 2 2 with s k = Φ k α k . ( 15
)
where ⊙ stands for entry-wise multiplication. Thus we are preventing the sparse model under construction from attempting to fit the invalid data. Other constraints can be easily imposed on the interpolated sparse components. For instance, in [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF], a total variation penalty is shown to enhance the recovery of piece-wise smooth components. Asking for the regularity across the gaps of some localized statistics ( e.g. enforcing that the empirical variance of a given inpainted sparse component be nearly equal outside and inside the masked areas) are other possible constraints. In practice, because of the lack of accuracy of some digital transformations we used in the spherical topology, additional constraints, which may be relaxed close to convergence, were also found useful in some cases to stabilize the described iterative algorithms.

It is proposed that a solution to the above minimization problem can be reached using the same iterative thresholding process as in the MCA algorithm detailed in the previous section, with the only required modification consisting in masking the full residual using M after each residual estimation. The MCA-inpainting algorithm is as follows :

1. Set the number of iterations Imax and the initial thresholds λ (0) 2. While λ (t) k is greater than a given lower bound λmin (e.g. can depend on the noise standard deviation), -Proceed with the following iteration to estimate components (s k ) k=1,...,K at iteration t:

For k = 1, • • • , K • Compute the residual term r (t) : r (t) = y - P k s(t-1) k • Estimate the current coefficients of s(t) k by thresholding with threshold λ (t) k : α(t) k = δ λ (t) k " T k " M ⊙ r (t) + s(t-1) k ""
• Get the new estimate of s k by reconstructing from the selected coeffcients

α(t) k : s(t) k = R k α(t)
k -Decrease the thresholds λ k following a given strategy.

The different thresholding strategies described in the previous section can be used in the proposed MCA inpainting iterative thresholding algorithm.

Example

A simple numerical experiment is shown on figure 5. Starting with a full satellite view of the Earth2 , an incomplete spherical map was obtained by randomly masking some of the pixels. In fact, as much as

Application in Astrophysics

A major issue in modern cosmology is the measurement and the statistical characterization (spatial power spectrum, Gaussianity) of the slight fluctuations in the Cosmic Microwave Background radiation field. These are indeed strongly related to the cosmological scenarios describing the properties and evolution of our Universe. Some 370 000 years after the 'Big Bang', when the temperature of the Universe was around 3000 K, thermal energy was no longer sufficient to keep electrons and positively charged particles apart so they combined. Photons were then set free in a nearly transparent Universe. Since the Universe further expanded, these photons are now in the microwave range but they should still be distributed according to a Black Body emission law. Indeed, before recombination, the Universe was a highly homogeneous opaque plasma in near thermal equilibrium in which photons and charged particles were highly interacting. Hence the slight fluctuations in matter density from which such large scale structures as galaxies or clusters of galaxies have evolved, are also imprinted on the distribution of photons. The Cosmic Microwave Background (CMB) was first observed in 1965 by Penzias and Wilson confirming a prediction made by Gamow in the late 1940's. But it was not until the early 1990's that evidence for small fluctuations in the CMB sky could finally be found thanks to the observations made by COBE [START_REF] Smoot | Structure in the COBE differential microwave radiometer first-year maps[END_REF]. This was confirmed by several subsequent observations and recently by NASA's Wilkin- The statistical analysis of this data set will help set tighter bounds on major cosmological parameters.

There are nonetheless a few practical issues and notably that several other astrophysical sources also emit radiation in the frequency range used for CMB observations [START_REF] Bouchet | Foregrounds and CMB experiments I. Semi-analytical estimates of contamination[END_REF]. Separating back the observed mixtures into maps of the different astrophysical contributions in order to isolate the CMB properly is a difficult inverse problem for which methods and algorithms are being actively designed (see e.g. [START_REF] Patanchon | Cosmic microwave background and foregrounds in Wilkinson Microwave Anisotropy Probe first-year data[END_REF][START_REF] Bobin | Morphological diversity and source separation[END_REF][START_REF] Moudden | Blind component separation in wavelet space: Application to CMB analysis[END_REF][START_REF] Pires | Sunyaev-Zel'dovich cluster reconstruction in multiband bolometer camera surveys[END_REF] and references therein). The estimated spherical CMB maps will inevitably be contaminated by some level of residual contributions, most significantly in the galactic region and at the locations of strong radio point sources. Therefore, it is common practice to mask out that part of the data (e.g. using the mask shown on figure 7 upper left, provided by the WMAP team) in order to reliably assess the non-gaussianity of the CMB field through estimated higher order statistics (e.g. skewness, kurtosis ) in various representations (e.g. wavelet, curvelet, etc.) [START_REF] Starck | Detecting cosmological non-gaussian signatures by multiscale methods[END_REF][START_REF] Jin | Cosmological non-gaussian signatures detection: Comparing performance of different statistical tests[END_REF]. But the gaps in the data thus created need to be handled properly as the detection of non-gaussianity in CMB would have a major scientific impact.

The proposed MCA-inpainting on the sphere was used here successfully to fill in the masked regions in order to restore the stationarity of the observed CMB field and lower the impact of the incompleteness of the data set on the estimated measures of non-gaussianity or any other non-local statistical test. The experiment was conducted on several simulations of full-sky Gaussian CMB maps. A typical CMB map (the CMB data map disclosed by the WMAP consortium) is shown on figure 6 along with the map obtained as a result of the inpainting process allowing for a first visual assessment of the quality of the proposed method. Figure 7 shows the wavelet decomposition of the inpainted map. We can see that the mask is not visible at all in the different scales. Here we have applied the MCA-Inpainting algorithm with 200 iterations and a single transform which was the Spherical Harmonic Decomposition. A more quantitative evaluation of the proposed inpainting algorithm is reported on figure 8 where plots of the estimated measures of non-Gaussianity on both the original map and the inpainted map are given. These reveal no significant discrepancy: we believe that the proposed method will help discriminate between truly non-Gaussian CMB and non-Gaussianity related to the non-stationarity of incomplete maps. This will be further investigated in the future.

Conclusion

This paper presented an extension of the Morphological Component Analysis method to the case of spherical maps. This was made possible thanks to the wealth of multiscale analysis tools and discrete transforms newly made available for the representation, analysis and synthesis of data on the sphere. An inpainting algorithm on the sphere was also described building on the ideas of MCA. The difficulties in porting MCA and the related inpainting algorithm to the spherical topology are less theoretical than practical. Indeed, the proposed algorithms are iterative and some of the digital transforms we used on the sphere were insufficiently accurate. As quickly mentioned, additional constraints were sometimes necessary to stabilize the convergence of the algorithms. However, lingering over such practicalities is not the purpose of this paper especially since they did not deeply affect the algorithms. We reported on a few applications of the proposed methods to challenging data analysis problems in physics and astrophysics. We expect these tools to be valuable in many other applications such as in areas where very smooth and precise motion is required ( e.g. moving chips on a conveyor belt in an ultra clean room in the semiconductor industry ) : hundred micron scale balls or ball bearings are used which must not have any imperfections on their surfaces above the nano-scale. Characterizing such tiny spherical objects with that kind of precision requires optical or X ray techniques which inherently produce artifacts. But since the morphology of the artifacts (interference pattern like repeated ring or arrays of light and dark regions) is far different than the bumps and scratches one is trying to avoid, they can be isolated using MCA techniques on the sphere and thus help assess the manufacturing process and make the changes required to meet specifications. Similar considerations are at play when treating ICF targets for laser fusion. The results reported in this paper allow us to expect that the described extensions of MCA and inpainting to the sphere will bear much fruit in the study of CMB, non-Gaussianity, and related matters such as astrophysical component separation [START_REF] Bobin | Morphological diversity and source separation[END_REF] in full-sky multichannel observations of the celestial sphere in the microwave range.

Figure 1 :

 1 Figure 1: Curvelets on the Sphere -The above figures show the backprojection of various curvelet coefficients at different scales and orientations on the sphere. Each map is obtained by setting all but one of the curvelet coefficients to zero, and applying an inverse curvelet transform. Depending on the scale and the position of the non zero curvelet coefficient, the reconstructed image presents a feature with a given width, length and orientation.

Figure 2 :

 2 Figure 2: Simple toy experiment with MCA on the sphere -The top map shows a linear combination of a spherical harmonic function and a localized Gaussian-like function on the sphere. The bottom maps show the resulting separated components that were obtained using the proposed Morphological Component Analysis on the sphere.

Figure 3 :

 3 Figure 3: left : Surface structures of ICF spherical shells measured on the nanometer scale are a superposition of global scale variations, isolated bumps and scratches as well as artifacts which look like interference patterns on intermediate scales. right : Coarsest scale of the undecimated isotropic wavelet transform of the surface measurements of an ICF target.

Figure 5 :

 5 Figure 5: Application of the proposed MCA-inpainting algorithm on the sphere. top left : original satellite view of the Earth ( mean = 76.9, σ = 47.7 ). top right : incomplete map retaining 40 percent of the original pixels. bottom left : inpainted map. bottom right : map of reconstruction errors ( mean = 0.0, σ = 2.86 empirically estimated from the reconstructed pixels only ).

Figure 6 :

 6 Figure 6: left : CMB data map provided by the WMAP team. Areas of significant foreground contamination in the galactic region and at the locations of strong radio point sources have been masked out. right : Map obtained by applying the proposed MCA-inpainting algorithm on the sphere to the former incomplete WMAP CMB data map.

Figure 7 :

 7 Figure 7: left : Mask provided by the WMAP team. The dark blue pixels indicate areas of high level foreground contamination in the WMAP CMB data map. From top to bottom and left to right : Maps of the wavelet decomposition on seven scales of the inpainted WMAP CMB map shown on the right of figure 6. From the visual point of view, the masked area cannot be distinguished anymore in the wavelet scales of the inpainted map.

Figure 8 :

 8 Figure 8: Horizontally is the scale number increasing for lower frequencies. Left : skewness of the wavelet coefficients in a given scale of the original complete simulated spherical CMB map (×) and of the inpainted map (♦). Right : kurtosis of the wavelet coefficients in a given scale of the original complete simulated spherical CMB map (×) and of the inpainted map (♦). Error bars were estimated on a small set of fifteen simulated complete CMB maps.

http://www.eso.org/science/healpix

availbale from : http://www.nasa.gov/vision/earth/features/bmng gallery 4.html

The WMAP data and mask we used here are available online at http://map.gsfc.nasa.gov/

http://astro.estec.esa.nl/Planck

Acknowledgments

Some of the results in this paper have been derived using the HEALPix (Górski, Hivon, and Wandelt 1999).


Noelia Do

Carmo Blanco

Julien Meyer
email: julien.meyer@gipsa-lab.fr

Michel Hoen

Fanny Meunier
email: fanny.meunier@unice.fr

Phoneme Resistance and Phoneme Confusion in Noise: Impact of Dyslexia

Keywords: word identification, speech-in-noise, dyslexia, phoneme resistance, phoneme confusion

Understanding speech in noisy environments is a challenge for almost everyone and particularly so for people with dyslexia. To better understand the phonological processing deficit, which has been posited as a core trait of dyslexia, we wanted to further characterize the impact of noise on speech perception. In this paper, we investigated phoneme resistance to noise for dyslexic and control adults and explored the pattern of errors produced by noise interference. Our aim was to examine differences between phoneme confusion matrices of the two populations.

Disyllabic nouns were embedded in noise and participants had to perform an auditory word identification task. Error rates, phoneme resistance and phoneme confusions were compared between a dyslexic and a group of matched controls.

Error rate was higher in the dyslexic group. However, no qualitative differences in the profile of errors were found. The coronals /ʃ and s/ were the most resistant phoneme in both groups while the labials /f, m and v/ were the most vulnerable. Although dyslexics showed a more scattered pattern of confusions, the matrices were correlated. Our results confirm a phonological deficit in dyslexia whereas they do not support the hypothesis of qualitative differences in phonological representation between the two groups.

Introduction

Dyslexia is a developmental language disorder occurring in the absence of sensory or neurological impairments. It is defined as a specific impairment in the development of normal reading skills. Although the etiology of dyslexia remains controversial, it is well established that people with dyslexia show deficits in phonological processing [START_REF] Ramus | Theories of developmental dyslexia: Insights from a multiple case study of dyslexic adults[END_REF]. Relatedly, it appears that dyslexia has effects not only on reading skills but also on the processing of acoustic and phonological information. In particular, it has been reported that an important proportion of dyslexics experience difficulties in understanding speech-in-noise [START_REF] Bradlow | Speaking Clearly for Children With Learning Disabilities[END_REF], in the absence of diagnosed hearing loss, while no differences have been observed in speech perception in quiet environments between dyslexics and control subjects [START_REF] Ziegler | Speech-perception-in-noise deficits in dyslexia[END_REF][START_REF] Brady | Speech perception and memory coding in relation to reading ability[END_REF]. This speech-innoise impairment has even been proposed as a core deficit of dyslexia as indeed, behavioral measures of speech-in-noise comprehension predict reading performances better than other cognitive, auditory or attentional measures [START_REF] Ziegler | Speech-perception-in-noise deficits in dyslexia[END_REF]. This impairment can be very disabling in daily life as most auditory communications take place in the presence of noise. The study of this deficit might shed light on the causes of the condition and might help in developing compensatory strategies.

The speech-in-noise deficit initially observed in dyslexic children is maintained at adult age and dyslexic adults also experience more difficulties understanding speech in noisy environments compared to normal-reading adults. With experience, dyslexics often develop compensatory strategies, giving sometimes the misleading impression that dyslexia disappears with age. In fact, it has been reported that on average, dyslexic adults performed less well than matched normal-readers at tasks evaluating speech perception in the presence of noise, even in experiments in which no written material was involved [START_REF] Dole | Speech-in-noise perception deficit in adults with dyslexia: Effects of background type and listening configuration[END_REF][START_REF] Ramus | Neuroimaging sheds new light on the phonological deficit in dyslexia[END_REF]. The first aim of this study is to replicate prior results that showed a deficit in perception of speech-in-noise in dyslexic adults. In a previous study, we looked at phoneme recognition error patterns for unimpaired participants [START_REF] Meyer | Speech Recognition in Natural Background Noise[END_REF]. The overall picture shows that vowels are much more resistant than consonants and that errors on consonants were mostly confusions, whereas deletion is also a type of errors for vowels. Performed analyses showed a limited impact of noise type on phoneme intelligibility. For example, the phoneme /v/, presented a stable pattern over noise conditions as being one of the less resistant consonants and /s/ and /ʃ/ the most resistant ones. Yet, the profile of phoneme errors resulting from the masking effect has never been studied for dyslexics. The second question we raise in this paper concerns differences between unimpaired subjects and dyslexics in the pattern of phoneme resistance and vulnerability to noise. It has been suggested that phonological representations are somehow degraded in dyslexia. The nature of the presumed degradation is still in debate. It includes, depending on authors, increased noise, phonetic underspecification, decreased temporal or spectral resolution, and/or less categorical representations (see [START_REF] Ramus | Neuroimaging sheds new light on the phonological deficit in dyslexia[END_REF] for an overview). If the phonological representations are degraded, we might expect fewer phonemes to be resistant to noise and more to be vulnerable.

The pattern of phoneme confusion could also shed light on phonological processing. It has been shown that confusion patterns are the result of not only acoustic properties and frequency of phonemes but also complex higher-level phonological processing [START_REF] Woods | Consonant identification in consonantvowel-consonant syllables in speech-spectrum noise[END_REF] and thus they can be asymmetric. For instance, /v/ elicits more /ð/ responses than /ð/ elicits /v/ [START_REF] Wang | Consonant confusions in noise: A study of perceptual features[END_REF]. It has been proposed that response biases contribute to confusion patterns. By characterizing the pattern of phoneme confusions in adults with dyslexia we can examine whether the response biases correspond to those present in non-dyslexic participants. If the phonological representations in dyslexia are degraded or less defined, we might expect a broader pattern of phoneme confusions under noise condition.

Methods

Participants performed a speech-in-noise identification task.

Participants

The study included 20 adults with a documented history of dyslexia (12 females, 8 males, mean age: 24.6 years, SD: 5.79) and 20 controls (16 females, 4 males, mean age: 24.35 years, SD: 5.68) matched in age, years of study, non-verbal intellectual abilities (as assessed with the Raven's Standard Progressive Matrices) and handedness. All dyslexic participants had been diagnosed by a psychologist or neuropsychologist prior to inclusion. All participants underwent a standardized reading test which showed significant differences between groups (see scores of the different tests and statistical analysis in [START_REF] Dole | Speech-in-noise perception deficit in adults with dyslexia: Effects of background type and listening configuration[END_REF]). All participants were right handed native French speakers, aged from 18 to 45 years, and had no other documented hearing or language disorders. The study was approved by the local ethics committee (ID RCB: 2008-A00708-47). Participants gave informed consent and received a monetary compensation. They were screened for audiometric pure-tone thresholds before participating in the study and no one exceeded 20 dB HL at any frequency between 250 and 8000 Hz.

Stimuli

Stimuli for the word recognition task consisted of 126 disyllabic words embedded in noise. The words were pronounced by a 24 year-old French native female speaker. They were recorded in a sound-proof room, at a sampling rate of 44.1 kHz. Stimuli were created by embedding one target-word in a randomly selected 4-s burst of noise, where the target words started 2.5s after noise onset.

Target words

The selected target words were disyllabic nouns with a middle range of frequency of occurrence (0.13 to 338.19 occurrence per million (opm), average: 16.82 opm, SD: 43.74) according to the French database Lexique2 [START_REF] New | Lexique 2: A new French lexical database[END_REF]. Regarding the first syllable, on which most of the analysis is focused, the set comprised 108 CV and 18 CCV (subsequently, the first consonant of the cluster will be named C1 and the second C1'). The occurrence of C1 phonemes was as follows: /p/ 18 times, /b,k/ 17 times, /ʁ/ 12 times, /ʃ/ 10 times /s,t,l/ 7 times, /d,f,g,m,v,ʒ/ 5 times and /n/ once.

Noise

Three different types of noise were used to avoid any specific effects: a cocktail-party noise (speech from a 4-talkers babble, 2 males and 2 females), a modulated noise and a stationary broadband noise (see [START_REF] Dole | Speech-in-noise perception deficit in adults with dyslexia: Effects of background type and listening configuration[END_REF] for a detailed description of the background noise and the different SNR conditions). Both, the modulated and the broadband noise were derived from the original cocktail-party sound and created by extracting the envelope information under 60 Hz.

Procedure

Participants were presented with the stimuli over headphones while seated in a sound-attenuated room. Only one target word out of the 126 was played in each trial and the order of presentation was random. Participants were instructed to listen carefully and to repeat the word they heard to the experimenter. Responses were transcribed verbatim by the experimenter and audiotaped to allow to double check the transcription. Participants did not receive feedback on their performance.

Analysis

To compare performance on speech understanding in noise between the dyslexic and the control group of participants we computed quantitative measures (i.e. error rates) as well as qualitative measures (i.e. profile of errors, including phoneme resistance to noise and phoneme confusions). Analyses of group differences in performance (excluding phoneme analysis) were assessed using unpaired t-tests and mixed model analysis of variance (ANOVA) with group (dyslexics and controls) as the between-subjects factor.

Greenhouse-Geisser corrected p-values were reported when the sphericity assumption was violated in repeated measures designs. Error bars in error rates and subsequent analysis were expressed as 95% bias-corrected and accelerated (BCa) bootstrap confidence interval (CI) of the mean computed with 1000 iterations. This method takes into account the real distribution of the sample. Cohen's d was used as the measure of effect size for t-tests.

Phoneme analysis

Since nearly 70% of the errors in both groups involved C1, phoneme resistance and phoneme confusions were computed only for C1. The phoneme /n/, which was presented in only one trial, was not included in these analyses. Phoneme error rates were calculated for each participant as the number of error for a particular phoneme related to the number of occurrences of the same phoneme. Statistical phoneme resistance to noise was assessed by comparing the percentage of error for each phoneme per participant to the phoneme percentage of occurrence. Thus by using one-sample two-sided t-tests we assessed if the percentage of error was higher or lower than expected or if there were no differences in resistance to noise. Scores in phoneme error rate were compared between groups with unpaired-sample t-tests. All the resulting p-values were false-discovery rate (fdr) corrected for multiple comparisons.

The reported phoneme confusion patterns include the total number of confusions between the studied French consonant phonemes in each group. Since the aim was to investigate differences between groups, the totals were not adjusted to the frequency of occurrence of each phoneme in the set of target words. For scale display purposes, we represented neither the number of times a phoneme was correctly reported nor the number of times a phoneme was omitted. Differences in phoneme confusions between the two groups were assessed with rank correlation analysis (Spearman's).

Results

The mean number of errors in word recognition in noise was higher in the dyslexic group (20.08%) than in the control group of participants (16.51%; See error rates and other descriptive measures of performance in Table 1). The difference in error rate between both groups (3.57%) was statistically significant as assessed by a two-sided unpaired t-test (see the analysis of group differences in Table 1), thus supporting a deficit in word identification in noise in dyslexia. Errors were either substitutions or deletions. The majority of them were substitutions in both groups. Substitution rates were significantly higher in the dyslexic group (17.1% vs. 14.0%), whereas omission rates were not (3.0% vs. 2.5%). When comparing the profile of errors we did not find differences in the percentage of errors corresponding to substitutions (about 85% in both groups).

We tested whether error rates were modulated by the position in the first syllable. Figure 1 shows error rate for each position in the first syllable and for each group. Vowels show the lowest error rate. A mixed-model ANOVA with position (C1, C1' and V1) as within-subjects factor and group (dyslexic and control participants) as between-subjects factor showed a main effect of position (F(2,76) = 47.62, p < .001, ηp 2 = .556). Post-hoc analysis indicated that error rate in C1' was higher than error rate in C1 and V1 (p = .01 and p < .001 and 95% CI of the difference [0.81-7.29 and 7.26-14.02] respectively). Similarly, error rate in C1 was higher than error rate in V1 (p < .001, [5.66-7.52]). Neither the main effect of group nor the interaction between group and position of error were significant (F(1,38) = 2.02, p = .164, ηp 2 = .050 and F(2,76) = 0.23, p = .665, ηp 2 = .006).

Phoneme resistance to noise

Despite the error rate being higher in C1', only 18 out of the 126 target disyllabic words were CCV. As a result, the majority of the observed errors in both groups consisted of C1 errors (the % of errors involving C1 was close to 70 %, as shown in Table 1). Only C1 was therefore considered to investigate phoneme resistance and confusion. Figure 2 depicts error rate on phoneme identification for each C1 phoneme and for each group. The C1 phonemes have been sorted by the error rate ranking of resistance to noise in the dyslexic group. Phoneme resistance to noise was investigated by comparing the percentage of error for each phoneme per participant to the phoneme percentage of occurrence. For the control group, the ranking of resistance to noise, from the most resistant to the most vulnerable, according to its error rate, was:

/s/> /ʃ/> /l/> /k/> /ʒ/> /ʁ/> /p/> /b/> /d/> /g/> /t/> /m/> /f/> /v/
And for the dyslexic group:

/ʃ/> /s/> /ʒ/> /k/> /d/> /l/> /ʁ/> /g/> /p/> /b/> /t/> /f/> /m/> /v/
For both groups, the phonemes whose errors of identification was significantly below chance level were: /s/, /ʃ/ and /k/ (p < fdr-adjusted critical p value of 0.014, t( 19 When assessing differences in phoneme resistance between the dyslexic and the control group, no significant results were found for any phoneme type (t < 1.4, p > .05).

Phoneme confusion matrices

Figure 3 shows the confusion matrices for the dyslexic group (A) and the control group (B). Phonemes have been sorted by the observed ranking of phoneme resistance in dyslexic adults. The most common phoneme confusion pairs were the labials /b/-/v/ and /v/-/b/, the plosives /p/-/b/ and the labials /m/-/b/. Although the confusion matrix in the dyslexic group is more scattered, the two matrices are highly correlated (Spearman's ρ = .735, p < .001, 95% BCa CI [.628-.826]).

Discussion

The aim of this study was to investigate the existence and the scope of the speech perception in noise deficit in adults with dyslexia regarding: a) performance and b) the profile of phoneme resistance to noise and phoneme confusion. We found that error rates were higher in the group with dyslexia. On the other hand, we were not able to show qualitative differences in the profile of phoneme errors.

Overall, the observed pattern of phoneme errors replicates previous findings in healthy adults. As expected, vowels were more resistant than consonants [START_REF] Meyer | Speech Recognition in Natural Background Noise[END_REF][START_REF] Varnet | Phoneme resistance during speech-in-speech comprehension[END_REF]. The phonemes /s/ and /ʃ/ were the most resistant consonants while /v/, /m/ and /f/ were the most vulnerable, similarly to what have been reported previously in French (see [START_REF] Meyer | Speech Recognition in Natural Background Noise[END_REF] for a detailed interpretation of this result). The pattern of confusions was also consistent with previous studies in French but also in other languages (see /b/-/v/ and /m/-/b/ confusions in English [START_REF] Woods | Consonant identification in consonantvowel-consonant syllables in speech-spectrum noise[END_REF] or /p/-/b/ in Spanish [START_REF] Moreno-Torres | Analysis of Spanish consonant recognition in 8-talker babble[END_REF]). Although we did not investigate all the French phonemes, it is not likely that differences are observed in the remaining C1 phonemes (i.e. /n/ and other less frequent C1 phonemes such as /x/, /j/, /z/).

Our data provide more behavioral evidence of a deficit in speech-in-noise perception in dyslexia using different type of stimuli. Compared to some previous studies [START_REF] Ziegler | Noise on, voicing off: Speech perception deficits in children with specific language impairment[END_REF][START_REF] Hazan | The Effect of Talker and Intonation Variability on Speech Perception in Noise in Children With Dyslexia[END_REF], the stimuli that were used had an ecological validity. Indeed, performance on discrimination between two phonemes or CV identification might overlook the deficit [START_REF] Hazan | Speech Perception Abilities of Adults With Dyslexia: Is There Any Evidence for a True Deficit?[END_REF]. Other studies that examined the perception of whole sentences under noise [START_REF] Bradlow | Speaking Clearly for Children With Learning Disabilities[END_REF] might allow the use of compensatory strategies. In our experiment we constrained context effects by using words presented in isolation and unrelated to each other.

In addition, we observed that the profile of errors for dyslexics is consistent with that of control participants. It has been proposed that phonological deficits in dyslexia might be due to poor phonological representation [START_REF] Boada | Deficient implicit phonological representations in children with dyslexia[END_REF][START_REF] Wagner | The nature of phonological processing and its causal role in the acquisition of reading skills[END_REF] and we hypothesized that if this was the case, the pattern of errors would be different between the two groups of participants. Yet, we were not able to show differences in phoneme vulnerability to noise or in phoneme confusions between groups. Our data adds to a growing literature supporting the hypothesis of an intact phonological representation in dyslexia [START_REF] Ramus | Neuroimaging sheds new light on the phonological deficit in dyslexia[END_REF][START_REF] Boets | Intact But Less Accessible Phonetic Representations in Adults with Dyslexia[END_REF][START_REF] Ramus | What phonological deficit?[END_REF]. Our study should be replicated in a children population to ensure that there is not a phonological deficit in dyslexic children, which could be compensated in adulthood.

Taken together, our data are in accordance with the external noise exclusion hypothesis [START_REF] Beattie | Dyslexic adults can learn from repeated stimulus presentation but have difficulties in excluding external noise[END_REF][START_REF] Sperling | Deficits in perceptual noise exclusion in developmental dyslexia[END_REF], which has been posited to account for reading and phonological deficits. It refers to the inability to exclude non-relevant noise. The quantitative differences observed between the two groups could be explained by the fact that adults with dyslexia have difficulties ignoring irrelevant background auditory information and therefore might have difficulties accessing phonological representations. This is in line with previous studies [START_REF] Ramus | Neuroimaging sheds new light on the phonological deficit in dyslexia[END_REF][START_REF] Ramus | What phonological deficit?[END_REF] that posited that phonological representations are less accessible in dyslexia. In particular, the access and use of phonological representations would be more difficult in certain situations that engage conscious awareness, verbal short-term memory, or speeded access [START_REF] Ramus | What phonological deficit?[END_REF]. Our study shows that noise addition may be one of these situations.

Conclusions

This study provides new behavioral data supporting a deficit in speech-in-noise recognition in adults with dyslexia. Moreover, we characterized, for the first time to our knowledge, the profile of phoneme resistance and phoneme confusion when perceiving speech-in-noise. The quantitative decline in performance does not seem to be accompanied by differences in the profile of phoneme resistance to noise or phoneme confusions. This supports the hypothesis that the phonological representations are preserved in dyslexia. The perceptual processing deficit in dyslexia might be related to the inability to exclude irrelevant information.

Figure 1 :

 1 Figure 1: Error rate for each position in the first syllable (C1' indicates the second consonant of the cluster). Error bars depict 95% BCa Bootstrap CI.

  ) = 13.44, d = 3.01, t(19) = 10.65, d = 2.38, t(19) = 2.81, d = 0.63 for the control group and t(19) = 6.78, d = 1.52 for /s/, t(19) = 13.31 d = 2.93 for /ʃ/and t(19) = 2.81, d = 0.63 for the dyslexic group respectively). On the contrary, identification rates were significantly below chance level for /f/, /m/ and /v/ for both groups (p < fdr-adjusted critical p value of 0.014, t(19) = 2.91, d = 0.65 for /f/, t(19) = 2.76, d = 0.62 for /m/ and t(19) = 3.44, d = 0.77 for /v/ for the control group of participants and t(19) = 2.84, d = 0.63 for /f/, t(19) = 2.69, d = 0.60 for /m/, t(19) = 2.85, d = 0.64 for /v/ for the dyslexic group).

Figure 2 :

 2 Figure 2: C1 phoneme error rate for each group.

Figure 3 :

 3 Figure 3: Confusion matrix for the dyslexic (A) and the control group of participants (B).

Table 1 :

 1 Descriptive and inferential statistics of performance. The values under the dyslexics and control group are expressed as mean (95% bias-corrected and accelerated bootstrap confidence interval).

	Measure	Dyslexics	Controls	t	df	p	95% CI diff Cohen's d
	Error rate	20.08 (18.20 -22.45)	16.51 (14.85-18.47)	2.43 38 .021	0.78-6.80	0.77
	Substitution rate	17.10 (15.25-19.27) 13.97 ( 12.55 -15.45) 2.62 38 .011	0.50 -5.55	0.83
	Omission rate	2.98 (1.89 -4.24)	2.54 (1.32 -3.90)	0.41 38 .693 -1.69 -2.42	0.13
	Errors profile
	% substitution within errors 86.15 (79.92 -91.30) 86.41 (79.00 -93.10) 0.057 38 .962 -9.17 -9.23	0.02
	% error involving C1	69.04 (64.31 -73.61) 69.67 (66.49 -72.25)	0.22 38 .824 -6.38 -5.25	0.06
	% errors involving C1'	12.88 (9.79 -16.00)	13.82 (10.77 -17.17)	0.41 38 .69	-5.51 -3.49	0.13
	% errors involving V1	32.52 (28.05 -37.10) 32.96 (26.94 -38.96)	0.11 38 .924 -8.33 -7.44	0.03

10.21437/Interspeech.2018-1271

Acknowledgements

We thank Nicolas Grimault for providing the background sound recordings and Marjorie Dole for her assistance on data collection. This research was supported by the European Research Council (starting-grant SpiN Project, n°209234).