Binbin Xu

Sabir Jacquir
email: sjacquir@u-bourgogne.fr

Gabriel Laurent

Jean-Marie Bilbault

Stéphane Binczak

Phase space reconstruction of an experimental model of cardiac field potential in normal and arrhythmic conditions

Cardiac arrhythmias are one of the most important death causes in the world. Compared to the numerical models, the experimental ones provide a more realistic tool to study the mechanisms of cardiac arrhythmias. The in vitro culture of cardiac cells developed on the Multi-Electrodes Array (MEA) constitutes a suitable model in this context. The extracellular field potential (EFP) acquired from the MEA can be used to measure the electrophysiological parameters of action potential. In this article, the stability of this experimental model is investigated using the phase space reconstruction in normal and in arrhythmia conditions. The results show that the embedding dimension of signal EFP changed slightly in both cases (normal conditions and arrhythmia). The parameter time lag τ in the normal conditions is lower than in the arrhythmia. The shape of attractors remains similar but disturbed in case of arrhythmia compared to the normal conditions.

Introduction

According to the recent report from the World Health Organization, the annual number of deaths due to cardiovascular disease will increase from 17 million in 2008 to 25 million in 2030 in the world. Among the death due to the cardiac problems, the arrhythmias play a major role, particularly in the atrial disorders. Facing to this challenge, many researches have been investigated in this field in order to understand the mechanism of these diseases. To achieve this goal, experimental models are much appreciated. They provide a more realistic situation than simulated ones. Ideally, it is better to conduct some in vivo studies in the heart. However, this is not always available and quite restricted by laws. Another serious problem of the in vivo studies is the heart beating. In order to overcome the limitations of the in vivo studies in the heart, the cardiac in vitro cultures are good alternatives and they permit even to study the arrhythmia at cellular level which could provide a better understanding of the mechanisms of arrhythmia generation and termination.

One of the technologies related to cardiac cultures is the MEA (Micro-Electrodes Array) which enables to monitor the culture for a longer time and is also less invasive compared to conventional electrophysiological methods as intracellular recording and patch-clamp. In a series of works [START_REF] Athias | Excitation spread in cardiac myocyte cultures using paired microelectrode and microelectrode array recordings[END_REF][START_REF] Jacquir | Paired microelectrodes and microelectrode array analysis of cardiac impulse propagation in cardiomyocyte cultures[END_REF], we use our platform MEA to study the electrical impulse propagation (extracellular field potential, EFP) in cardiomyocytes culture under basal conditions. The results showed that in normal conditions, the electrical activities in the culture can be represented by the propagation of planar waves (action potential). In later studies [START_REF] Jacquir | Investigation of micro spiral waves at cellular level using a microelectrode array technology[END_REF][START_REF] Athias | In vitro simulation of spiral waves in cardiomyocyte networks using multi-electrode array technology[END_REF], cardiac arrhythmias (spiral waves, SW) have been induced in the cardiomyocytes culture by electrical stimulations, in agreement with results in [START_REF] Kondratyev | Dynamic changes of cardiac conduction during rapid pacing[END_REF] which showed that a rapid stimulation could alter cardiac conduction and induce then arrhythmias. These results showed that this experimental model provide an excellent tool to study the cardiac arrhythmias itself and the effect of electrical stimulation on it. In order to study the behavior and the stability of the experimental model, non-linear methods are applied to the acquired signals. As the physiological signals, generally acquired as time series, are often marked by significant non-linearity that conventional analysis methods often fail to identify. It is more suitable to use non-linear methods. One of the popular nonlinear methods is phase space reconstruction, which is a valuable tool for the studies of this kind of signals. For a dynamical system, the phase space consists of a set of points (every point corresponds to a system state) which forms trajectories of the system. It shows information such as the existence of an attractor for selected parameter values. So this paper is aimed to determine the system dimension of this experimental model and to characterize it by reconstructing its phase space, in two cases: normal condition and under electrical stimulations.

The paper is organized as follows. The Section 2 presents the cardiomyocytes culture preparing and the electrical stimulation to the culture to obtain arrhythmic phenomenon. The system dimension determination and the reconstruction of phase space are discussed in Section 3.

Experimental models and system

Extracellular field potentials

The MEA consists of 60 electrodes aligned in a matrix form 8 × 8 with an inter-electrodes distance of 100 µm. The planar electrodes have a diameter of 30 µm. The cultures from cardiac muscular cells were developed directly on the MEA [START_REF] Athias | Development of cardiac physiopathological models from cultured cardiomyocytes[END_REF]. Every acquired signal is in fact a local extracellular field potential (EFP), since it comes from several cells around the electrode. They correspond to the combination the 1st and 2nd order derivative of the action potential. In basal conditions, the EFP signals are quite regular (Fig. 1).

Figure 1: EFP in basal conditions. Every panel corresponding to an acquisition of 8 s from the real position of an electrode. (sampling frequency 500 Hz)

Arrhythmic signals in cardiomyocytes culture induced by electrical stimulation

To induce arrhythmia in heart or in cardiomyocytes culture, there exist generally two methods : by injection of specific drugs such as aconitine and acetylcholine etc. [START_REF] Allessie | Experimental evaluation of moe's multiple wavelet hypothesis of atrial fibrillation[END_REF] and by vagal or electrical stimulation. We are interested in the method of electrical stimulation as rapid pacing with low energy can induce arrhythmia and promote it in isolated heart or in cardiac cultures [START_REF] Jacquir | Investigation of micro spiral waves at cellular level using a microelectrode array technology[END_REF][START_REF] Athias | In vitro simulation of spiral waves in cardiomyocyte networks using multi-electrode array technology[END_REF]. In fact, these electrical stimulations can also be a potential method of atrial defibrillation. In recent years, the concept of sub-threshold stimulation with low amplitude and high frequency becomes more and more known and accepted [START_REF] Osipov | Controlled movement and suppression of spiral waves in excitable media[END_REF][START_REF] Shajahan | Spiral-wave turbulence and its control in the presence of inhomogeneities in four mathematical models of cardiac tissue[END_REF][START_REF] Xu | A hybrid stimulation strategy for suppression of spiral waves in cardiac tissue[END_REF]. In order to study experimentally the sub-threshold defibrillation, it is better to understand firstly the culture model itself under electrical stimulation. By electrical stimulation, cardiac arrhythmia can be induced in cardiac cultures (Fig. 2).

Phase space reconstruction

The principle of phase space reconstruction is to transform the properties of a time series into topological properties of a geometrical object which is embedded in a space, wherein all possible states of the system are represented, each state corresponds to a unique point, and this reconstructed space sharing the same topological properties as the original space. Mathematically, the states of a m dimensional dynamic system can only be characterized by m independent quantities. Such a set of m independent quantities represents the coordinates of the phase space (more exactly: the coordinates in a m dimensional space).

One of the most used methods of phase space reconstruction is the time delay embedding. Since this method does not require that the treated system could be mathematically defined, explicitly, it fits well with 1D time series.

X(m, τ) = [x(t), x(t + τ), . . . , x(t + τ(m -1))],t = 0, 1, 2, . . .
where X denotes the system states of time series x(t), a function of its embedding dimension m and time lag τ. The reconstruction preserves the properties of the dynamic system that do not change under coordinate changes, that is to say, it keeps only the topological aspect instead of the geometric shape in phase space. To better present these signals in their phase space, the m and τ should be determined very carefully. This is the purpose of the following paragraphs.

Embedding dimension m

As previously mentioned, a well-defined embedding dimension is essential to a phase space. For experimental data, we do not unfortunately know a priori the dimension of the studied system. The phase space reconstruction is based on the principle of connecting the state points to create attractor trajectories of the system, if m is not large enough, the states points on the trajectories would be very close, there will be intersections between the trajectories of the attractor. A higher value of m, theoretically, is possible by just creating a larger space that contains minimal space. However, this will cause not only redundant problems, but also an efficiency issue, as it requires a much important and unnecessary computation power. The most common method used to estimate the embedding dimension is the method of False Nearest Neighbor (FNN) [START_REF] Kennel | Determining embedding dimension for phase-space reconstruction using a geometrical construction[END_REF]. The criterion for choosing the value of m corresponds to the minimum value of m for which the FNN pass into is close to zero.

By the method on FNN, it is found that in normal condition and in case of arrhythmia, the embedding dimension of EFP depends on signals itself (Figures ??). The obtained values are all in the range of embedding dimension reported by others studies for cardiac signals [START_REF] Fell | Nonlinear analysis of continuous ECG during sleep I[END_REF]. Even m changed slightly, it remains globally stable, which implies that the model is robust and stable.

Time lag τ

Non-linear variables change of a dynamic system will act as bases of coordinates. Ideally, this database should be independent of selected time lag τ. The goal is to find the largest value of τ so that the resulting coordinates are relatively independent. If τ is too small, the trajectories of x(t) and x(t + τ) close to one another, could even be indistinguishable and it will create redundant coordinates. For this reason, the basic criterion for choosing τ is that x(t) and x(t + τ) should be independent. On the other hand, if τ is sufficiently too large, the two coordinates are statistically independent. This implies that the trajectories of attractor projected on the two axes are not correlated, which makes the reconstruction phase space useless. Without a priori knowledge of the system, especially for experimental data, it is not obvious to properly determine this parameter. The method by autocorrelation function is used here [START_REF] Albano | Singular-value decomposition and the grassbergerprocaccia algorithm[END_REF]. According to correlation theorem , it is known that under certain conditions two observations of the same process (x(t 1 ), x(t 2 )) or two functions (x 1 (t), x 2 (t)) to describe this process may be identical, which means x(t 1 ) = x(t 2 ) or x 1 (t) = x 2 (t). Basically, the autocorrelation function C(τ) defines the similarity of a signal to itself but temporally shifted by τ which depends on the properties of treated data, necessary always to examine it by several methods in order to obtain a better estimation. Taking the signals that are previously studied, the estimation results of τ are presented in Fig. 5. By comparing these signals, we find that the values of τ for arrhythmic signals (39 ± 8) is higher than the ones for normal signals 22 ± 6. τ may serve as an indication of arrhythmia.

Phase space reconstruction

The reconstructed phase space are presented in Fig. 6 for regular signals and in Fig. 7 in case of cardiac arrhythmia. As shown in Fig. 6, the trajectories of the normal signals illustrate the existence of an attractor in their phase spaces. The global forms of these trajectories are rather triangular; the widths of the trajectories are relatively narrow, which means that the trajectories are stable. In case of arrhythmias (Fig. 7), the reconstructed phase space become very disturbed and are no longer regular. The differences are remarkable. However, the same attractors as the ones for regular signals are more or less found. Actually, if one compares the regular signals and arrhythmic signals, despite some variations in amplitude and periods, arrhythmic signals can be considered as "stable".

x(t) x(t -24) n˚5 x(t) x(t -20) n˚15 x(t) x(t -25) n˚20 x(t) x(t -27) n˚29

Conclusion

In our previous work [START_REF] Jacquir | Investigation of micro spiral waves at cellular level using a microelectrode array technology[END_REF], inducing cardiac arrhythmia by electrical stimulation is confirmed by our experimental studies. It showed by the resulting period-doubling route to chaos that in case of arrhythmia, EFP signals become chaotic. To reveal more information embedded in the time series, nonlin- ear methods for physiological signals are more suitable than linear ones.

x(t) x(t -38) n˚5 x(t) x(t -35) n˚15 x(t) x(t -38) n˚20 x(t) x(t -39) n˚29
In case of arrhythmias induced by electrical stimulations, the embedding dimension of EFP signals changed slightly, but generally stable, which is a sign of the model stability. However, for the parameter "time lag" τ, it is in opposite situation. For arrhythmic signals, the values of their τ are increased. So the change from normal situation to arrhythmia could be identified by m and τ, which would provide potential index for arrhythmia detection / identification.

The experimental model can be considered as "stable" : after the electrical stimulation, the global trajectories of EFP signals in the reconstructed phase space remain similar. In normal conditions, the signals are regular, their phase spaces have a triangular form, and the widths of the trajectories are relatively narrow. "Attractors" are formed at the upper part in the phase space. With the electrical stimulation, the trajectories of the model are very disturbed. They became much wider. However, their forms are more or less the same as in normal conditions; the global triangular form is kept, which reflected that the system dynamics of this model is generally stable.

In conclusion, the cardiomyocytes cultures are quite stable. The arrhythmic signals can be easily identified by time delay τ. The electrical stimulation does not modify the global dynamics of the experimental model, which makes it be a robust tool for studying the mechanism of cardiac arrhythmia and the defibrillation by sub-threshold electrical stimulation.

Figure 2 :

 2 Figure 2: Arrhythmic signals after electrical stimulations.

Figure 3 :

 3 Figure 3: Estimation of m by FNN, regular signals' examples, electrodes 11 and 54.

Figure 4 :

 4 Figure 4: Estimation of m by FNN, arrhythmic signals.

Figure 5 :

 5 Figure 5: Comparison of τ by autocorrelation function (first 1/e acrossing) for two type of EFP signals: regular signals and arrhythmic ones. τ is used as samples (sampling frequency 500 Hz).

Figure 6 :

 6 Figure 6: Phase space reconstruction for regular signals (from electrodes 5, 15, 20 and 29).As shown in Fig.6, the trajectories of the normal signals illustrate the existence of an attractor in their phase spaces. The global forms of these trajectories are rather triangular; the widths of the trajectories are relatively narrow, which means that the trajectories are stable. In case of arrhythmias (Fig.7), the reconstructed phase space become very disturbed and are no longer regular. The differences are remarkable. However, the same attractors as the ones for regular signals are more or less found. Actually, if one compares the regular signals and arrhythmic signals, despite some variations in amplitude and periods, arrhythmic signals can be considered as "stable".

Figure 7 :

 7 Figure 7: Phase space reconstruction for arrhythmic signals (from electrodes 5, 15, 20 and 29).

Acknowledgements

B. Xu is financially supported by the French IHU in cardiac rhythmology L'Institut de rythmologie et modélisation cardiaque LIRYC and the regional council of Aquitaine.


P Paykari

J.-L Starck

M J Fadili

True CMB Power Spectrum Estimation

Keywords: Cosmology : Cosmic Microwave Background, Methods : Data Analysis, Methods : Statistical

Context.

Aims. The cosmic microwave background (CMB) power spectrum is a powerful cosmological probe as it entails almost all the statistical information of the CMB perturbations. Having access to only one sky, the CMB power spectrum measured by our experiments is only a realization of the true underlying angular power spectrum. In this paper we aim to recover the true underlying CMB power spectrum from the one realization that we have without a need to know the cosmological parameters. Methods. The sparsity of the CMB power spectrum is first investigated in two dictionaries; Discrete Cosine Transform (DCT) and Wavelet Transform (WT). The CMB power spectrum can be recovered with only a few percentage of the coefficients in both of these dictionaries and hence is very compressible in these dictionaries.

Results. We study the performance of these dictionaries in smoothing a set of simulated power spectra. Based on this, we develop a technique that estimates the true underlying CMB power spectrum from data, i.e. without a need to know the cosmological parameters. Conclusions. This smooth estimated spectrum can be used to simulate CMB maps with similar properties to the true CMB simulations with the correct cosmological parameters. This allows us to make Monte Carlo simulations in a given project, without having to know the cosmological parameters. The developed IDL code, TOUSI, for Theoretical pOwer spectrUm using Sparse estImation, will be released with the next version of ISAP.

Introduction

Measurements of the CMB anisotropies are powerful cosmological probes. In the currently favored cosmological model, with the nearly Gaussian-distributed curvature perturbations, almost all the statistical information are contained in the CMB angular power spectrum. The observed quantity on the sky is generally the CMB temperature anisotropy Θ( p) in direction p, which is described as T ( p) = T CM B [1 + Θ( p)]. This field is expanded on the spherical harmonic functions as

Θ( p) = +∞ ℓ=0 ℓ m=-ℓ a[ℓ, m]Y ℓm ( p) , (1)
where a[ℓ, m] =

S 2 Θ( p)Y * ℓm ( p)d p , (2)
S 2 ⊂ R 3 is the unit sphere, ℓ is the multipole moment which is related to the angular size on the sky as ℓ ∼ 180 • /θ and m is the phase ranging from -ℓ to ℓ. The a[ℓ, m] are the spherical harmonic coefficients of the (noise-free) observed sky.

For a Gaussian random field, the mean and covariance are sufficient statistics, meaning that they carry all the statistical information of the field. In case where the random field has zero mean, E(a 00 ) = 0 and the expansion can be started at ℓ = 2, neglecting the dipole terms, i.e. ℓ = 1 1 . For ℓ 2, the triangular array (a[ℓ, m]) ℓ,m represents zero-mean, complex-valued random coefficients, with variance

E(|a[ℓ, m]| 2 ) = C[ℓ] > 0 , (3)
⋆ paniez.paykari@cea.fr 1 The dipole anisotropy is dominated by the Earth's motion in space and it is hence ignored.

where C[ℓ] is the CMB angular power spectrum, which only depends on ℓ due the isotropy assumption. Therefore, from (3), an unbiased estimator of C[ℓ] is given by the empirical power spectrum

C[ℓ] = 1 2ℓ + 1 m |a[ℓ, m]| 2 . (4)
Furthermore, as the random field is stationary, the spherical harmonic coefficients are uncorrelated,

E(a[ℓ, m]a * [ℓ ′ , m ′ ]) = δ ℓℓ ′ δ mm ′ C[ℓ] . (5)
Since they are Gaussian they are also independent. The angular power spectrum depends on the cosmological parameters through an angular transfer function T ℓ (k) as

C[ℓ] = 4π dk k T 2 ℓ (k)P (k) , (6)
where k defines the scale and P (k) is the primordial matter power spectrum.

Making accurate measurements of this power spectrum has been one of the main goals of cosmology in the past two decades. We have seen a range of ground-and balloon-based experiments, such as Acbar (Reichardt et al. 2009) and CBI (Readhead et al. 2004), as well as satellite experiments, such as WMAP (Bennett et al. 2003) and the recently launched satellite Planck (Planck Collaboration et al. 2011). All these experiments estimate the CMB angular power spectrum from a sky map, which is a realization of the underlying true power spectrum; no matter how much the experiments improve, we are still limited to an accuracy within the cosmic variance. This means that even if we had a perfect experiment (i.e. with zero instrumental noise) we would not be able to recover a perfect power spectrum due to the cosmic variance limit.

In this paper we investigate the possibility of estimating the true underlying power spectrum from a realized spectrum; an estimation of the true power spectrum without a need to know the cosmological parameters. For this we exploit the sparsity properties of the CMB power spectrum, and capitalize on it to propose an estimator of the theoretical power spectrum. This estimate will not belong to a set of possible theoretical power spectra (i.e. all C[ℓ] that can be obtained by CAMB2 by varying the cosmological parameters). Instead, such an estimation should be useful for other applications, such as:

• Monte Carlo: we may want to make Monte Carlo simulations in some applications without assuming the cosmological parameters. • Wiener filtering: Wiener filtering is often used to filter the CMB map and it requires the theoretical power spectrum as an input. We may not want to assume any cosmology at this stage of the processing. • Some estimators (weak lensing, ISW, etc.) require the theoretical power spectrum to be known. Using a data-based estimation of the theoretical C[ℓ] could be an interesting alternative, or at least a good first guess in an iterative scheme where the theoretical C[ℓ] is required to determine the cosmological parameters.

Sparsity of the CMB Power Spectrum

A brief tour of sparsity

A signal X = (X[1], . . . , X[N ]) considered as a vector in R N , is said to be sparse if most of its entries are equal to zero. If k number of the N samples are not equal to zero, where k ≪ N , then the signal is said to be k-sparse. In the case where only a few of the entries have large values and the rest are zero or close to zero the signal is said to be weakly sparse (or compressible). With a slight abuse of terminology, in the sequel, we will call compressible signals sparse. Generally signals are not sparse in direct space, but can be sparsified by transforming them to another domain. For example, sin(x) is 1-sparse in the Fourier domain, while it is clearly not sparse in the original one. In the so-called sparsity synthesis model, a signal can be represented as the linear expansion

X = Φα = T i=1 φ i α[i] , (7)
where α[i] are the synthesis coefficients of X, Φ = (φ 1 , . . . , φ T ) is the dictionary, and φ i are called the atoms (elementary waveforms) of the dictionary Φ. In the language of linear algebra, the dictionary Φ is a N × T matrix whose columns are the atoms normalized, supposed here to be normalized to a unit

ℓ 2 -norm, i.e. ∀i ∈ [1, T ], φ i 2 2 = N n=1 |φ i [n]| 2 = 1 3
. A function can be decomposed in many dictionaries, but the best dictionary is the one with the sparsest (most economical) representation of the signal. In practice, it is convenient to use dictionaries with fast implicit transform (such as Fourier transform, wavelet transform, etc.) which allow us to directly obtain the coefficients and reconstruct the signal from these coefficients using fast algorithms running in linear or almost linear time (unlike matrix-vector multiplications). The Fourier, wavelet and discrete cosine transforms provide certainly the most well known dictionaries. A comprehensive account on sparsity and its applications can be found in the monograph [START_REF] Starck | Sparse Image and Signal Processing[END_REF].

2.2. Which Dictionary for the Theoretical CMB Power Spectrum?

We investigate the sparsity of the CMB power spectrum in two different dictionaries, both having a fast implicit transform: the Wavelet Transform (WT) and the Discrete Cosine Transform (DCT). Fig. 1. A theoretical CMB power spectrum along with the reconstructed power spectra, using the DCT and WT dictionaries. The panels show the reconstructions for different fractions of the coefficients used. The inner plots show the differences between the actual and the reconstructed power spectra. Both dictionaries suffer from boundary effects, but this is more severe for DCT as the corresponding atoms are not compactly supported. It is worth mentioning that the power spectrum that is decomposed onto the two dictionaries is in the form ℓ(ℓ + 1)C[ℓ]/2π.

Figure 1 shows an angular power spectrum (calculated by CAMB (Lewis et al. 2000) with WMAP7 (Larson et al. 2010) parameters) along with the DCT-and WT-reconstructed power spectra with a varying fraction of the largest transform coefficients retained in the reconstruction. The inner plots show the difference between the actual power spectrum and the reconstructed ones. It can be seen that with only a few percentage of the coefficients the shape of the power spectrum is correctly reconstructed in both dictionaries. The height and the position of the peaks and troughs are of great importance here as the estimation of the cosmological parameters heavily relies on these characteristics of the power spectrum. The best domain would be the one with the sparsest representation and yet the most accurate representation of the power spectrum. Let C[ℓ] (M ) be its best M -term approximation, i.e. obtained by reconstructing from the M -largest (in magnitude) coefficients of C[ℓ] in a given domain. To compare the WT and DCT dictionaries, we plot the resulting non-linear approximation (NLA) error curve in Figure 2, which shows the reconstruction error E M as a function of M , the number of retained coefficients;

E M = C[ℓ] -C[ℓ] (M ) 2 C[ℓ] 2 × 100 . (8)
As M increases we get closer to the complete reconstruction and the error reaches 0 when all the coefficients have been used. Usually the domain with the steepest E M curve is the sparsest domain. In this case though both dictionaries Fig. 2. Non-Linear Approximation (NLA) error curves for the two dictionaries. Below 1% the DCT curve is dropping faster, which means it is doing a better job. However, past ∼ 2% the DCT curve flattens off while WT decreases to ∼ 0 very quickly.

have very similar behaviors. There is only a small window in the coefficients for which DCT does a better job than WT. However, DCT flattens after using ∼ 1% of the coefficients and does not improve the reconstruction until a big proportion of the coefficients have been used. Both dictionaries seem to suffer from boundary issues at low and high ℓs. This can be solved for high ℓs as one can always perform the reconstruction beyond the desired ℓ. For low ℓs it can be solved by different means, such as extrapolation of the spectrum. Note that the boundary issues are more severe in the DCT domain than WT; this is due to the fact that DCT atoms are not compactly supported.

Next we investigate the sparsity of a set of realized spectra in the two dictionaries. We simulate 100 maps from the theoretical power spectrum used previously and estimate their power spectra using equation 4 . As before, we decompose each realization in the DCT and WT dictionaries and reconstruct keeping increasing fractions of the largest coefficients. At this stage, it is important to note that, as we are dealing with the empirical power spectrum, we are no longer in an approximation setting but rather in an estimation one. Indeed, the empirical power spectrum can be seen as a noisy version of the true one. Intuitively, reconstructing from a very small fraction of high coefficients will reject most of the noise (low estimator variance) but at the price of retaining only a small fraction of the true spectrum coefficients (large bias). The converse is true when a large proportion of coefficients is kept in the reconstruction. Therefore, there will exist a threshold value that will entail a bias-variance tradeoff, hence minimizing the estimation risk. This is exactly the idea underlying thresholding estimators in sparsifying domains.

This discussion is clearly illustrated by the inner plots of Figure 3, which shows the normalized mean-square error (NMSE) defined as

NMSE M = C[ℓ] -C[ℓ] (M ) 2 C[ℓ] 2 × 100 , (9)
as a function of the fraction of coefficients used in the reconstruction. The error is large when only a few coefficients are used. As more coefficients are included, one starts to recover the main (i.e. the general shape of the spectrum) features of Fig. 3. A simulated CMB power spectrum along with the reconstructed spectra, using the DCT and WT dictionaries. The black solid line is the true underlying power spectrum from which the simulations were made. The blue and red dots show the simulated and the reconstructed power spectra respectively. With only 1% of the coefficients, DCT can recover the input power spectrum (i.e. the black solid line) very well, recovering the peaks and troughs accurately. Unlike DCT, WT seems to have difficulties in recovering the peaks and troughs. The inner plots shows the NMSE curves.

the power spectrum. With more coefficients, more noise enters the estimation and the error increases again. The NMSE curve shows a clear minimum at which the underlying true power spectrum is best recovered. Despite the differences in the performance of the two dictionaries, the minima of the NMSE are around the same proportions of the coefficients. This is because the NMSE reflects a global behavior. On the one hand, although the DCT can recover the features of the spectrum correctly, it is less smooth than WT. Conversely, the WT cannot reconstruct the proper shape of the power spectrum, but provides a smoother estimate.

To summarize, from the above discussion, we conclude the following:

• the CMB power spectrum is very sparse in both the DCT and WT dictionaries, although their sparsifying capabilities are different; • DCT recovers global features of spectrum (i.e. the peaks and troughs) while WT recovers localized features;

• in the case of realizations, WT recovers more localized (noisy) features than the global ones, while the DCT concentrates on the global features.

In the next section, these complementary capabilities of the DCT and WT transforms will be combined to propose a versatile way for adaptively estimating the theoretical power spectrum from a single realization of it.

Sparse Reconstruction of the Theoretical Power Spectrum

Let's start with the simple model where the observed signal Y is contaminated by a zero-mean white Gaussian noise, Y = X + ε, where X is the signal of interest and ε ∼ N (0, σ 2 ). Sparse recovery with an analysis-type sparsity prior amounts to finding the solution of the following problem:

min X Φ T X 1 s.t. Y -X 2 ≤ δ , (10)
where Φ T X represent the transform coefficients of X in the dictionary Φ, and δ controls the fidelity to the data and obviously depends on the noise standard deviation σ.

Let's now turn to denoising the power spectrum from one empirical realization of it. In this case, however, the noise is highly non-Gaussian and needs to be treated differently. Indeed, as we will see in the next section, the empirical power spectrum will entail a multiplicative χ 2 -distributed noise with a number of degrees of freedom that depends on ℓ. That is, the noise has a variance profile that dependents both on the true spectrum and ℓ. We therefore need to stabilize the noise on the empirical power spectrum prior to estimation, using a Variance Stabilization Transform (VST). Hopefully, the latter will yield stabilized samples that have (asymptotically) constant variance, say 1, irrespective of the value of the input noise level.

Variance Stabilizing Transform

In the statistical literature the problem of removing the noise from an empirical power spectrum goes by the name of periodogram denoising (Donoho 1993). In (Komm et al. 1999), approximating the noise with a correlated Gaussian noise model, a threshold was derived at each wavelet scale using the MAD (Median of Absolute Deviation) estimator. A more elegant approach was proposed in (Donoho 1993;Moulin 1994), where the so-called Wahba VST was used. This VST is defined as:

T (X) = (log X + γ) √ 6 π , (11)
where γ = 0.57721... is the Euler-Mascheroni constant. After the VST, the stabilized samples can be treated as if the noise contaminating them were white Gaussian noise with unit variance. We will take a similar path here, generalizing the above approach to the case of the angular power spectrum. Indeed, from (4), one can show that under mild regularity assumptions on the true power spectrum, 12), it is appealing then to take the logarithm so as to transform the multiplicative noise Z into an additive one. The resulting log-stabilized empirical power spectrum reads

C[ℓ] d → C[ℓ]Z[ℓ], where ∀ℓ ≥ 2, 2LZ[ℓ] ∼ χ 2 2L , L = 2ℓ + 1 . ( 12
) d → means convergence in distribution. From (
C s [ℓ] := T ℓ ( C[ℓ]) = log C[ℓ] -µ L = log C[ℓ] + η[ℓ] . (13)
where

η[ℓ] := log Z[ℓ] -µ L , L = 2ℓ + 1.
Using the asymptotic results from (Bartlett & Kendall 1946) on the moments of log -χ 2 variables, it can be shown that µ

L = ψ 0 (L) -log L, E(η[ℓ]) = 0 and σ 2 L = Var [η[ℓ]] = ψ 1 (L)
, where ψ m (t) is the standard polygamma function, ψ m (t) = d m+1 dt m+1 log Γ(t). We can now consider the stabilized C s [ℓ] as noisy versions of the log C[ℓ], where the noise is zero-mean additive and independent. Owing to the Central Limit Theorem, the noise tends to Gaussian with variance σ 2 L as ℓ increases. At low ℓ, normality is only an approximation. In fact, it can be show that the noise η[ℓ] has a probability density function of the form

p η (ℓ) = (2L) L 2 L Γ(L) exp L ℓ + µ L -e ℓ+µ L , (14)
which might be used to estimate the thresholds in the wavelet domain.

In order to standardize the noise, the VST (13) will be slightly modified to the normalized form

C s [ℓ] := T ℓ ( C[ℓ]) = log C[ℓ] -µ L σ L = X s [ℓ] + ε[ℓ] . ( 15
)
where now the noise ε[ℓ] is zero-mean (asymptotically) Gaussian with unit variance, and

X s [ℓ] := log C[ℓ]/σ L .
It can be checked that the Wahba VST ( 11) is a specialization of ( 15) to L = 0.

In the following, we will use the operator notation T (X) for the VST that applies (15) entry-wise to each X[ℓ], and R(X) its inverse operator, i.e. R(X) :

= R ℓ (X[ℓ]) ℓ with R ℓ (X[ℓ]) = exp(σ L X[ℓ]).

Signal detection in the wavelet domain

Without of loss of generality, we restrict our description here to the wavelet transform. The same approach applies to other sparsifying transforms, e.g. DCT, just as well.

In order to estimate the true CMB power spectrum from the wavelet transform, it is important to detect the wavelet coefficients which are "significant", i.e. the wavelet coefficients which have an absolute value too large to be due to noise (cosmic variance + instrumental noise). Let w j [ℓ] the wavelet coefficient of a signal Y at scale j and location ℓ. We define the multiresolution support M of Y as:

M j [ℓ] = 1 if w j [ℓ] is significant, 0 if w j [ℓ] otherwise. ( 16
)
For Gaussian noise, it is easy to derive an estimation of the noise standard deviation σ j at scale j from the noise standard deviation, which can be evaluated with good accuracy in an automated way (Starck & Murtagh 1998). To detect the significant wavelet coefficients, it suffices to compare the wavelet coefficients in magnitude |w j [ℓ]| to a threshold level t j . This threshold is generally taken to be equal to κσ j , where κ ranges from 3 to 5. This means that a small magnitude compared to the threshold implies that the coefficients is very likely to be due to noise and hence insignificant. Such a decision rule corresponds to the hard-thresholding operator

if |w j [ℓ]| ≥ t j then w j [ℓ] is significant , if |w j [ℓ]| < t j then w j [ℓ] is not significant. ( 17
)
To summarize, The multiresolution support is obtained from the signal Y by computing the forward transform coefficients, applying hard thresholding, and recording the coordinates of the retained coefficients.

Power Spectrum Recovery Algorithm

Let's now turn to the adaptive estimator of the true CMB power spectrum C[ℓ] from its empirical estimate C[ℓ]. As we benefit from the (asymptotic) normality of the noise in the stabilized samples C s [ℓ] in (15), we are in position to easily construct the multiresolution support M of C s as described in the previous section. Once the support M of significant coefficients has been determined, our goal is reconstruct an estimate X of the true power spectrum, known to be sparsely represented in some dictionary Φ(regularization), such that the significant transform coefficients of its stabilized version reproduce those of C s (fidelity to data). Furthermore, as a power spectrum is a positive, a positivity constraint must be imposed. These requirements can be cast as seeking an estimate that solves the following constrained optimization problem:

min X Φ T X 1 s.t. X 0 M ⊙ Φ T T (X) = M ⊙ Φ T C s , (18)
where ⊙ stands for the Hadamard product (i.e. entry-wise multiplication) of two vectors. This problem has a global minimizer which is bounded. However, beside non-smoothness of the l 1 -norm and the constraints, the problem is also non-convex because of the VST operator T . It is therefore far from obvious to solve.

In this paper we propose the following scheme which starts with an initial guess of the power spectrum X (0) = 0, and then iterates for n = 0 to N max -1,

X = R T X (n) + ΦM ⊙ Φ T C s -T X (n) X (n+1) = P + Φ ST λn (Φ T X) , (19)
where P + denotes the projection on the positive orthant and guarantees non-negativity of the spectrum estimator, ST λn (w) = (ST λn (w[i])) i is the soft-thresholding with threshold λ n that applies term-by-term the shrinkage rule

ST λn (w[i]) = sign(w[i])(|w[i]| -λ n ) if |w[i]| λ n , 0 otherwise . (20)
Here, we have chosen a decreasing threshold with the iteration number n, λ n = (N maxn)/(N max -1). More details pertaining to this algorithm can be found in [START_REF] Starck | Sparse Image and Signal Processing[END_REF]

Instrumental Noise

In practice, the data are generally contaminated by an instrumental noise, and estimating the true CMB power spectrum C[ℓ] from the empirical power spectrum C[ℓ] requires to remove this instrumental noise. The instrumental noise is assumed stationary and independent from the CMB. We will also suppose that we have access to the power spectrum of the noise, or we can compute the empirical power spectrum S N [ℓ] of at least one realization, either from a JackKnife data map or from realistic instrumental noise simulations. The above algorithm can be adapted to handle this case after rewriting the optimizing problem as follows:

min X Φ T X 1 s.t. X 0 M ⊙ Φ T T (X + S N ) = M ⊙ Φ T C s . ( 21
)
Thus, (19) becomes

X = R T X (n) + S N + ΦM ⊙ Φ T C s -T X (n) + S N -S N X (n+1) = P + Φ ST λn (Φ T X) . (22)

Combining Several Dictionnaries

We have seen in Section 2.2 that the WT and DCT dictionaries had complementary benefits. Indeed each dictionary is able to capture well features with shapes similar to its atoms. More generally, assume that we have D dictionaries Φ 1 , • • • , Φ D . Given a candidate signal Y , we can derive a support M d associated to each dictionary Φ d , for d ∈ {1, • • • , D}.

The optimization problem to solve now reads min

X Φ T X 1 s.t. X 0 M d ⊙ Φ T d T (X + S N ) = M d ⊙ Φ T d C s , d ∈ {1, • • • , D} . (23)
Again, this is a challenging optimization problem. We propose to attack it by applying successively and alternatively (22) on each dictionary Φ d . Algorithm 1 describes in detail the different steps. Estimation 4: Initialize X (0) = 0, 5: for n = 0 to Nmax -1 do 6:

Z d = X (n) . 7: for d = 1 to D do 8: e Z = R " T " Z d + b SN " + Φ d M ⊙ " Φ d T " C s -T " Z d + b SN """" -b SN . 9: Z d+1 = P+ " Φ d ST λn (Φ d T e Z) " . 10:
end for 11:

X (n+1) = Z D+1 . 12: λn+1 = Nmax-(n+1)
Nmax-1 . 13: end for 14: Get the estimate e X = X (Nmax) .

Data with Instrumental Noise

Here we present the performance of the TOUSI algorithm in the presence of instrumental noise. The noise maps were simulated using a theoretical (PLANCK level) noise power spectrum. They were added to the CMB maps simulated previously and the power spectra of the combined maps were estimated using equation 4.

Figure 4 shows the reconstruction of the theoretical CMB spectrum in the presence of noise. The blue dots show the empirical power spectrum of one realization having instrumental noise. Yellow dots show the estimated power spectrum of one of the simulated noise maps. Green dots show the the spectrum with the noise power spectrum removed. The black and red solid lines are the input and reconstructed power spectra respectively. The theoretical power spectrum can be reconstructed up to the point where the structure of the power spectrum has not been destroyed by the instrumental noise. In our case, having PLANCK level noise, this goes to ℓ up to 2500. It can be seen that TOUSI can do a great job in reconstructing the input power spectrum even in the presence of instrumental noise.

Sparsity versus Averaging

A very common approach to reduce the noise on the power spectrum is the moving average filter, i.e. average values in a given window,

C A [b] = 1 b(b + 1)ω b b+ ω b 2 ℓ=b- ω b 2 ℓ(ℓ + 1) C[ℓ] , (24)
and the window size ω b is increasing with ℓ. Here, we use window sizes of {1, 2, 5, 10, 20, 50, 100} respectively for ℓ ranging from {2, 11, 31, 151, 421, 1201, 2501} to {10, 30, 150, 420, 1200, 2500, 3200}, which have also been used in the framework of the PLANCK project in (Leach et al. 2008).

Fig. 4. Power spectrum estimation in the presence of instrumental noise. The blue dots show the empirical power spectrum of one realization having instrumental noise. Yellow dots show the estimated power spectrum of one of the simulated noise maps. Green dots show the the spectrum with the noise power spectrum removed. The black and red solid lines are the input and reconstructed power spectra respectively. The inner plots show a zoomed-in version.

Figure 5 shows the average error the 100 realizations as a function of ℓ

E[ℓ] = 1 100 100 i=1 C[ℓ] -C i [ℓ] 2 , ( 25
)
where C i is the estimated power spectrum from the i-th realization. We display the errors for the spectra estimated by the empirical estimator (the realization, black dotted line), the averaging estimator (red dashed line) and TOUSI (solid blue line). The cosmic variance is over-plotted as a solid black line. We can see that the expected error is highly reduced when using the sparsity-based estimator.

Conclusion

Measurements of the CMB anisotropies are powerful cosmological probes. In the currently favored cosmological model, with the nearly Gaussian-distributed curvature perturbations, almost all the statistical information are contained in the CMB angular power spectrum. In this paper we have investigated the sparsity of the CMB power spectrum in two dictionaries; DCT and WT. In both dictionaries the CMB power spectrum can be recovered with only a few percentages of the coefficients, meaning the spectrum is very sparse. The two dictionaries have different characteristics and can accommodate reconstructing different features of the spectra; The DCT can help recover the global features of the spectrum, while WT helps recover small localized features. The sparsity of the CMB spectrum in these two domains has helped us develop an algorithm, TOUSI, that estimates the true underlying power spectrum from a given realized spectrum. This algorithm uses the sparsity of the CMB power spectrum in both WT and DCT domains and takes the best from both worlds to get a highly accurate estimate from a single realization of the CMB power spectrum. This could be a replacement for CAMB in cases where knowing the cosmological parameters is not necessary. The developed IDL code will be released with the next version of ISAP (Interactive Sparse astronomical data Analysis Packages) via the web site: http://jstarck.free.fr/isap.html

Algorithm 1 :

 1 TOUSI Power Spectrum Smoothing with D dictionaries Require: Empirical power spectrum b C, D dictionaries Φ1, ..., ΦD, noise power spectrum b SN , Number of iterations Nmax, Threshold κ (default value is 5). Detection 1: Compute C s using (15). 2: For all d, compute the decomposition coefficients W d of C s in Φ d , W d = Φ T d C s . 3: For all d, compute the support M d from W d with the threshold κ, assuming standard additive white Gaussian noise.

Fig. 5 .

 5 Fig. 5. Mean error for the 100 realizations, for the realizations (black dotted line), the averaging denoising (red dashed line) and the sparse wavelet filtering (blue solid line). The inner plot shows a zoom between l = 2000 and l = 3000.

CAMB solves the Boltzmann equations for a cosmological model set out by the given cosmological parameters.

The lp-norm of a vector X, p ≥ 1, is defined asX p = `Pi |X[i]| p ´1/p , with the usual adaptation X ∞ = maxi X[i].

Acknowledgments

The authors would like to thank Marian Douspis, Olivier Doré and Amir Hajian for useful discussions. This work is supported by the European Research Council grant SparseAstro (ERC-228261)


Ciara R Wigham

Review of 'Contemporary Computer-Assisted Language Learning'

Wigham, C.R. (à paraître, juin 2013). Review of Contemporary Computer-Assisted Language Learning, Language Learning and Technology.

REVIEW OF CONTEMPORARY COMPUTER-ASSISTED LANGUAGE LEARNING

Book reference: M. Thomas, H. Reinders & M. Warschauer (Eds.) (2013). Contemporary Computer-Assisted Language Learning. Bloomsbury: London & New York.

Contemporary Computer-Assisted Language Learning, edited by Michael Thomas, Hayo Reinders, and Mark Warschauer, maps the parameters of contemporary computer-assisted language learning (CALL) and illustrates how a range of different areas, as well as research approaches, are shaping this field. The introductory chapter states four objectives fulfilled by the volume. Firstly, it offers an overview of the historical perspectives of CALL and identifies relationships with other disciplines. Secondly, it provides a critical synopsis of CALL research and ways in which this empirical base is leading to richer theoretical understandings. Thirdly, the introduction documents research that explores relationships between theory and practice in a range of educational settings, and finally, it outlines new research directions and approaches.

The edited volume offers a balanced representation of voices through its 18 chapters contributed by researchers in Canada, Germany, Hong Kong, Japan, Spain, the United Kingdom, the United States, and Venezuela. A wide range of topics are covered, which supports the editors' observation that CALL has developed from a narrow area of specialist interest to an area of scholarship in its own right, with independent subfields, including teacher education, evaluation, language testing, social media, and task design.

The contributions are organized in a three-part architecture: CALL context, CALL learning environments, and CALL in language education. The volume features an insightful foreword by Mike Levy, who examines reasons to employ the label CALL. He describes the book as essential reading for postgraduate students of language teaching, for CALL researchers-in short, for -old hands and newcomers alike‖ (p. xviii).

In Part I, The CALL Context, five chapters examine the historical development of the field and its research trends, in view of theories of language education and the process of technological integration and normalization.

In Chapter 2, the late Graham Davies and his colleagues Sue Otto and Bernd Rüschoff provide a historical backdrop for the overall volume. Following a discussion on the origin of the term CALL, the authors describe how CALL has been shaped, in a linear fashion by the technologies available, and in a multipronged fashion, by trends in SLA and other language education theories. In examining the history of CALL from the pioneer PLATO project of the 1960s to the social turn of the 2000s, the chapter matches technological developments with pedagogical and methodological progress in language learning.

Chapter 3 discusses CALL research trends and outlines their enrichment by methodological approaches from other disciplines. Carla Meskill and Joy Quah review techniques employed in studies of language learning with online social media. Firstly, they examine studies that consider that an online environment impacts in unique ways on communicative exchanges. How researchers develop and approach their datasets according to the theoretical underpinnings and purpose of their studies is exemplified. Secondly, datasets related to selfempowerment, community building and learner confidence are examined. Special methodologies are described, including learner self-reporting in diary data and class blogs, which allows for the investigation of social-affective dimensions of online social media. Finally, methodological approaches in studies that concern teaching practices are considered. Meskill and Quah conclude that CALL research trends are moving beyond learner interaction datasets, in large part due to new uses of technology that inspire new enquiry methodologies.

In Chapter 4, Gary Motteram, Diane Slaouti, and Zeynep Onat-Stelma suggest that CALL teacher education be considered in light of sociocultural theory and educational perspectives rather than only in SLA. The contributors use a number of up-to-date references to describe ways in which educational technology and the study of learner viewpoints and teacher beliefs have impacted teacher education. The authors then detail sociocultural theory and relate it to activity theory in order to represent -all of the different elements of a situation that need to be taken into account when we consider … teacher education‖ (p. 64). Applications of sociocultural theory are exemplified through an exploration of various teaching practices of language instructors from around the world. The data illustrate how teachers' beliefs about technology use relate to decision-making processes about technology in the classroom and how such beliefs can lead to different language learning opportunities.

The volume then turns to the subfield of language testing. James Dean Brown proposes an examination of past developments, current trends, and future perspectives in computer-based language testing (CBLT). He examines changes in CBLT that have taken place since his own synthesis over ten years ago [START_REF] Brown | Computers in language testing: Present research and some future directions[END_REF]. In particular, he details the move away from practical studies that address topics including item banking, computer-adaptive language testing, and the effectiveness of computers in testing, to studies that provide overviews of CBLT literature and that consider learners' experiences of taking CBL tests and teacher-training needs. This chapter offers a rich acronym glossary and an interesting reflection on the extent to which earlier predictions about CBLT research directions have been met.

The final chapter in Part I focuses on material design. Mirjam Hauck and Sylvia Warnecke forward that social presences (SPs) have important implications for material and task design. After examining the concept of SPs the researchers suggest the need for studies into SPs to consider learners' perspectives and not solely those of teachers and researchers. They illustrate this by focusing on an EAP tutor training course which employed asynchronous discussion forums. The study shows how course tasks, including collective reflections on issues including participation and motivation, produced and enriched SPs. This leads to the suggestions that SP be reconsidered in light of the Community Indicators Framework and that studies focusing on multimodal, rather than textual environments are required.

Part II is entitled CALL Learning Environments, which the editors say will be used to examine technologies that -go beyond the classroom in isolation… [to help] broaden students' access to learning opportunities‖ (p. 119).

The contribution by Robert O'Dowd focuses on telecollaboration. He reviews models and practices for engaging language learners in virtual contact to develop language and intercultural skills. In Chapter 9, Randall Sadler and Melinda Dooly focus on virtual worlds. Their chapter is clearly divided into two sections. The first characterizes social virtual worlds and outlines their historical development, before providing a research review. The authors trace the history of these environments back to the PLATO project of the 1960s and describe how virtual worlds are aimed at particular age groups. They detail research studies grouped around three themes: collaboration, learner anxiety, and the representation of virtual world users as avatars. They then highlight the need for further studies into how these environments are being used for language learning, which leads directly to their second section focused on this topic. An ethnographic case study details a project that used Second Life to investigate project-based language learning approaches in primary education. The data analysis focuses on recurrent patterns of language use that are situated in response to the virtual world context.

Chun Lai, Ruhui Ni, and Yong Zhao, in Chapter 10, review three types of commercial digital games used in language learning: video games, massive multi-player role playing-games, and 3D collaborative virtual environments. The authors discuss the different affordances of each game type and the pedagogical considerations that teachers account for when preparing learners for gameplay and when connecting the latter with curriculum objectives. Emphasis is placed on scaffolding gameplay, debriefing gaming sessions, and designing tasks. The chapter then turns to the need to balance entertainment and learning in serious educational language learning game design. They illustrate this discussion with the example of Zon, a MMORG for Chinese language learning. They discuss lessons learned by the game's designers, which include fitting the game into the instructional context, enhancing its adaptability to the context and restructuring it to meet institutional needs.

In Chapter 11, Glenn Stockwell gives a balanced review of mobile-assisted language learning (MALL). After situating MALL with relation to integrated CALL and the concept of digital natives, Stockwell outlines the perceived advantages for mobile learning and describes ways that research studies have focused on podcasting using MP3 players, vocabulary learning and audio guides using PDAs, and vocabulary tasks using mobile phones. The chapter details physical, physiological and pedagogical issues of concern with mobile learning and the consequences for language learning. Issues related to small screen size, assumptions that learners can use the devices and the impact of students' non-pedagogical use of devices on their pedagogical use are discussed. Finally, future directions for MALL are detailed, including the inclusion of GPS technologies and the interaction possibilities between learners and everyday objects. Stockwell concludes that the biggest challenge for MALL concerns how to combine physical spaces and virtual environments.

Whilst the previous chapters focus on sophisticated technologies that can foster learning in new contexts, in Chapter 12 Dafne Gonzalez and Rubena St. Louis offer a complementary study of CALL in low-tech contexts. The chapter compares factors that describe low-tech contexts in CALL literature with survey responses from language teachers worldwide. The aim of this study is to determine the factors for a low-tech context, describe the manner in which resources and tools are used in these contexts, and give practical examples of strategies used to overcome obstacles faced. The results show that low-tech contexts can be defined, first in terms of insufficient access to ICT (physical constraints), and second in terms of human constraints, whereby teachers are unable to use resources adequately. The number of recommendations given for using CALL in these environments is rich, and the authors situate their survey results with reference to a wide literature selection.

In Part III of the work, CALL in Language Education, seven chapters examine specialized areas of teaching and research that have stemmed from the diversity of CALL.

Mathias Schulze and Trude Heift's contribution concerns the developments in intelligent CALL (ICALL).

The chapter offers two perspectives. Firstly, by concentrating on SLA theories concerning interaction and noticing, the authors suggest what ICALL resources offer to language teachers. They focus on projects concerning vocabulary and grammar acquisition, and on writing and reading comprehension. Their historically informed analysis details concrete examples including the Robo-Sensei, FLUENT and QuickAssist projects. Secondly, the researchers focus on three approaches for using corpora in ICALL. They illustrate how both learner and reference corpora are used to evaluate NLP tools and aid their design. They also examine the application of NLP tools and corpora to CALL environments.

Chapter 14 focuses on computer-supported reading. Firstly, Youngmin Park, Binbin Zheng, Joshua Lawrence and Mark Warschauer discuss the components of word decoding, language comprehension and text interpretation that are central to second language reading and review research in these areas. Secondly, they detail two approaches to using digital media to support the second language reading process: visual-syntactic text formatting and collaborative blogging. In the first approach, presenting texts in cascaded versions helps compensate for second language learners' limitations in syntactic knowledge and aids reading comprehension.

The second example illustrates how blogging helped engage young learners of English and developed their sense of audience and authorship. The chapter concludes by presenting and exemplifying a typology of how digital resources can be used to enhance reading comprehension.

The volume then turns to writing skills. In Chapter 15, Volker Hegelheimer and Jooyoung Lee discuss classroom use of technology-assisted writing tools. After outlining six tips for teaching writing with CALL, the authors detail approaches to using automated essay and collaborative writing Web 2.0 environments that have the potential to shift focus in writing instruction from form to meaning. Each technology is considered in terms of current practice, its advantages and disadvantages, and future research agendas. The researchers provide readers with a discussion of considerations for the attention of teachers involved in teaching writing with technology.

CALL and less commonly taught languages (LCTL) are at the center of Chapter 16. Richard Robin, firstly, discusses how to define a LCTL in terms of the characteristics that are -subject to the offerings and challenges posed by current technology‖ (p. 304). The features of language level difficulty, proficiency expectations, and small Internet footprints are outlined. Robin then turns to resources the Internet may offer to LCTLs. He describes the increased availability of pedagogical materials, particularly pertaining to business language, the possibility for the creation of modular learning materials, and the opportunities offered to for self-study. Although this discussion is exemplified throughout, the chapter then focuses on the tools Google offers and other resources for authentic listening. Following a discussion of the issue of representing non-Roman writing systems across computing platforms and the difficulties learners face in producing such scripts on screen, the chapter concludes that distance and hybrid courses could help save, if not expand, LCTL teaching.

Paige Ware and Greg Kessler take up the theme of digital feedback within CALL in Chapter 17. Rather than provide a strict definition of feedback, the authors explore three of its facets: mode of delivery, strategies for delivery, and focus on feedback provided. They then examine digital feedback in CALL on writing.

Computer-generated feedback that largely concerns error correction is discussed, as well as electronically delivered peer feedback focusing on idea generation or form. The researchers subsequently explore digital feedback on speaking. They discuss the use of automated speech recognition to provide pronunciation pointers to students and recent technologies that allow for human-generated digital speech interaction. The chapter concludes by specifying that a future research agenda for this area should focus upon what constitutes effective feedback and quality across the three different facets of feedback.

In the penultimate chapter, Michael Thomas explores the relationships between task-based language teaching (TBLT) and CALL. Whilst these two fields were perceived, until the 1990s, as running in parallel, Thomas provides a historical outline of ways in which TBLT may inform CALL. He insists on the potential pedagogical implications relating to designing, sequencing and understanding tasks, as well as learners' behavior in a task cycle. A case study of two EFL learners at a Japanese university is then presented to explore collaborative technologies. The study draws on a virtual world and examines learner identity and learner challenges in using technology with authentic tasks. The author describes how the learners developed their L2 identities in ways -that could feedback positively into their academic learning contexts‖ (p. 355) but underlines an overreliance on their L1 for preparative and collaborative stages of the tasks.

To close the volume, Hayo Reinders and Philip Hubbard address learner autonomy. Their contribution offers a review of learner autonomy in language learning and teaching, before turning to the potential affordances offered and constraints posed by technology for learner autonomy development. The contributors structure these two sections using the same sub-headings, which offers the reader a balanced synthesis. The areas discussed include access, storage and retrieval, sharing and recycling of materials and situated learning. The researchers then focus upon ways in which the constraints discussed may be overcome. This revolves around the need to train learners to efficiently use technology-mediated materials, to guide them to make appropriate choices concerning materials and task selection, to encourage them to invest in communities of practice and to develop specific initiatives that target advancing learner autonomy.

As readers who have made it to this point of the review of Contemporary Computer-Assisted Language Learning will tell, the breadth and depth of the chosen topics for this book is impressive. Topics are treated from historical, theoretical, technological and practical perspectives in a very comprehensive manner. We must hope that university libraries are willing to add the work to their collections. Although proceeds from the book are to be donated to the Japanese Red Cross, the price risks impacting the number of students willing to invest in the resource independently.

Readers will appreciate the way in which they are carefully guided through the three different parts of the book by section introductions that clearly map out the contributions included and demonstrate the interest of each. However, at times more cross-references between the individual chapters would have been useful, and the lack of a concluding chapter by the editors to tie the three parts together is noticeable. The major strength of this volume is undoubtedly the way in which it takes a broad look at the evolution of CALL into a field of research and practice within its own right, whilst accounting for many of the key aspects that constitute the field in such a detailed manner.

  She also discriminates between distance and hybrid learning and discusses the relationship of distance learning with DCALL and CALL. The concepts of flexibility and explicitness are central to her proposed model of teaching and learning in supported distance learning, a model which is built around a learning design approach and the presence of institutional and distributed learning strategies. The model is applied to examples from distance language learning literature. The conclusion discusses issues that currently occupy DCALL researchers. These issues emerge from a corpus of 54 DCALL research articles and include multimodality, knowing more about learners and distance language teacher training.

The chapter describes how telecollaboration has moved from text-based, asynchronous configurations to multimodal exchanges. O'Dowd focuses on research findings from telecollaborative projects in the areas of linguistic proficiency, learner autonomy and intercultural learning. He details ways in which qualitative studies have focused on project design and structure. Throughout the chapter, he outlines the complications of integrating telecollaboration into formal education but offers examples of initiatives that help to normalize this process; Connect and Erasmus programmes are highlighted as well as other curricular initiatives to integrate telecollaboration into language teacher training.

Marie-Noëlle Lamy offers the next contribution that distinguishes CALL from distance CALL (DCALL). Lamy outlines why the fields are often considered as overlapping, due to a lack of dedicated dissemination territories for DCALL researchers and terminological confusions related to distance education. She addresses this confusion by clarifying several terms, including flexible learning, open learning, supported learning and e-learning.


F Lecellier

S Jehan-Besson

J Fadili

G Aubert

M Revenu

Optimization of divergences within the exponential family for image segmentation

In this work, we propose novel results for the optimization of divergences within the framework of region-based active contours. We focus on parametric statistical models where the region descriptor is chosen as the probability density function (pdf) of an image feature (e.g. intensity) inside the region and the pdf belongs to the exponential family. The optimization of divergences appears as a flexible tool for segmentation with and without intensity prior. As far as segmentation without reference is concerned, we aim at maximizing the discrepancy between the pdf of the inside region and the pdf of the outside region. Moreover, since the optimization framework is performed within the exponential family, we can cope with difficult segmentation problems including various noise models (Gaussian, Rayleigh, Poisson, Bernoulli ...). We also experimentally show that the maximisation of the KL divergence offers interesting properties compare to some other data terms (e.g. minimization of the anti-log-likelihood). Experimental results on medical images (brain MRI, contrast echocardiography) confirm the applicability of this general setting.

Optimization of divergences between pdfs : general setting

In this section, we set our general setting for segmentation through the optimization of distances between pdfs or more generally divergences.

General setting

Consider a function y : R n → χ ⊂ R which describes the feature of interest. The term y(x) then represents the value of the feature y at location x where x ∈ R n . Let q(y, Ω)

Introduction

We propose here to focus on the segmentation of homogeneous regions in noisy images using statistical region-based active contour models (RBAC). In RBAC, region-based terms can be advantageously combined with boundary-based ones [START_REF] Kass | Snakes : Active contour models[END_REF][START_REF] Caselles | Geodesic active contours[END_REF]. The evolution equation is generally deduced from a general criterion to minimize that includes both region integrals and boundary integrals. The combination of those two terms in the energy functional allows the use of photometric image properties, such as texture [START_REF] Aujol | Wavelet-based level set evolution for classification of textured images[END_REF] and noise [START_REF] Martin | Influence of the noise model on level set active contour segmentation[END_REF], as well as geometric properties such as the shape prior of the object to be segmented. In statistical region-based active contours, see [START_REF] Cremers | A review of statistical approaches to level set segmentation: integrating color, texture, motion and shape[END_REF] for a review, image features (e.g. intensity) are considered as random variables whose distribution may be parametric (e.g. Gaussian) or non parametric [START_REF] Aubert | Image segmentation using active contours: Calculus of variations or shape gradients ?[END_REF]. Classically, the authors consider the minimization of the anti-log-likelihood for segmentation [START_REF] Zhu | Region competition: unifying snakes, region growing, and bayes/MDL for multiband image segmentation[END_REF][START_REF] Paragios | Geodesic active regions: A new paradigm to deal with frame partition problems in computer vision[END_REF][START_REF] Martin | Influence of the noise model on level set active contour segmentation[END_REF]. In this paper, we rather focus on the optimization of distance between pdfs. Such distances or more generally divergences can be used in two different manner. On the one hand, they can be used for segmentation with distribution intensity prior and in this case, we aim at minimizing the distance between the pdf of the evolving region and a reference one. On the other hand, they can be used for segmentation without reference and in this second case, we aim at maximizing the distance between the pdf of the inside region and the pdf of the outside region. In the literature, the minimization of divergences between non parametric pdfs has first been proposed in [START_REF] Aubert | Image segmentation using active contours: Calculus of variations or shape gradients ?[END_REF] for video sequences. It has then been developed for cardiac structures tracking in perfusion MRI (p-MRI) sequences in [START_REF] Rougon | Region-based statistical segmentation using informational active contours[END_REF]. As far as segmentation using the maximization of divergences is concerned, some authors [START_REF] Michailovich | Image segmentation using active contours driven by the bhattacharyya gradient flow[END_REF] have also proposed to take benefit of the maximization of the Bhattacharya distance of non parametric pdfs for segmentation. On the other hand, divergences between Gaussian distributions have been developed for DTI segmentation in [START_REF] Wang | DTI segmentation using an information theoretic tensor dissimilarity measure[END_REF].

In this paper, we propose to set a general framework for the optimization of divergences between parametric pdfs within the exponential family. To the best of our knowledge, such a framework has never been studied for region-based active contour segmentation. The rationale behind using the exponential family is that it includes, among others, Gaussian, Rayleigh, Poisson and Bernoulli distributions that have proven to be useful to model the noise structure [START_REF] Martin | Influence of the noise model on level set active contour segmentation[END_REF] in many real image acquisition devices (e.g. Poisson for photon counting devices such as X-ray or CCD cameras, Rayleigh for ultrasound images, etc). Using shape derivative tools as in [START_REF] Jehan-Besson | DREAM 2 S: Deformable regions driven by an eulerian accurate minimization method for image and video segmentation[END_REF][START_REF] Aubert | Image segmentation using active contours: Calculus of variations or shape gradients ?[END_REF], our effort focuses on constructing a general expression for the derivative of the energy (with respect to a domain), and on deriving the corresponding evolution speed. Our general framework is also specialized to some particular cases, such as the optimization of the Kullback-Leibler (KL) divergence [START_REF] Kullback | Information Theory and Statistics[END_REF], which gives a simple expression of the derivative. This theoretical framework is then more explicitly detailed and illustrated for the case of the segmentation without reference. In this case, we aim at maximizing the dissimilarity between the pdf of the intensity within the region inside the evolving contour and the pdf of the intensity within the region outside the contour. In other words, we perform a competition between the pdfs of these two regions through the maximisation of divergences. Experimental results are given for the particular case of the KL divergence. We experimentally compare this data term to the classical minimization of the anti-log-likelihood [START_REF] Zhu | Region competition: unifying snakes, region growing, and bayes/MDL for multiband image segmentation[END_REF][START_REF] Chan | Active contour without edges[END_REF] for the segmentation of the White Matter in brain MRI and we show that KL maximisation is able to extract a single Gaussian from a mixture of Gaussian. We also show the applicability of our data term for the segmentation of the left ventricle in contrast echocardiography where the noise is modelled using Rayleigh.

In this paper, we first set our general setting and introduce shape gradients in section 2. In section 3, we propose to give some general results for the exponential family and then for the shape derivative of divergences between pdfs. These results are then specialized for the KL divergence using the Maximum Likelihood Estimation (MLE) for the parameters. Experimental results for the maximization of KL divergence are given in section 4. be the probability density function (pdf) of the feature y within the image region of interest. We now assume that we have a function Ψ : R + × R + → R + which allows us to compare two pdfs. This function is small if the pdfs are similar and large otherwise. It allows us to introduce the following functional which represents the distance or more generally the divergence between the current pdf estimate q(y, Ω) and another one p(y) which may also depend on another domain:

D(Ω) = Z χ Ψ (q(y, Ω), p(y)) dy.
(

) 1
The distance can be for example the symmetrized Kullback-Leibler divergence when Ψ(q, p) = 1 2 (p(y) log p(y) q(y,Ω) + q(y, Ω) log q(y,Ω) p(y) ). Such divergences represent a general setting for both segmentation with and without reference. Indeed, in segmentation problems, we generally search for homogeneous regions regarding with a given feature. We may then modelize the segmentation problem as the maximization of the distance between the pdf of the feature within the inside region and the pdf of the feature within the outside region. In order to fix ideas, let us consider a partition of an image in two regions where Ω is the inside region and Ω c the complementary outside region. The segmentation may then be formulated as the maximization of the following criterion:

D(Ω, Ω c ) = Z χ Ψ (q(y, Ω), p(y, Ω c )) dy.
(

) 2
On the other hand, we can also consider that we have a reference histogram p re f and that we search for the domain that minimizes the divergence between q and p re f . This last framework may be applied to tracking or to supervised segmentation where a reference pdf is learned on the region of interest. The theoretical results given in this paper can be used for both applications.

Shape gradient descent

In order to find an optimum, we perform a shape gradient descent using region-based active contours. We then have to compute the derivative of the criterion according to the domain using shape derivation tools [START_REF] Delfour | Shape and geometries[END_REF]. Shape derivative tools applied to region-based active contours are described in [START_REF] Jehan-Besson | DREAM 2 S: Deformable regions driven by an eulerian accurate minimization method for image and video segmentation[END_REF][START_REF] Aubert | Image segmentation using active contours: Calculus of variations or shape gradients ?[END_REF] and we won't remind all the definitions in this paper. Let us just remind that, from the shape derivative, we can derive the evolution equation that will drive the active contour towards a (local) minimum of the criterion. Let us suppose that the shape derivative of the criterion D(Ω) in the direction V may be written as follows:

< D (Ω), V >= - Z ∂Ω speed(x, Ω)(V(x) • N(x))da(x) , ( 3
)
where N is the unit inward normal to ∂Ω and da its area element. When minimizing the distance D(Ω), interpreting equation (3) as the L 2 inner product on the space of velocities, the straightforward choice is to take V = speed(x, Ω)N.

When minimizing the distance D(Ω), we can then deduce the following evolution equation:

∂Γ ∂τ = speed(x, Ω) N(x) , (4)
On the contrary, when maximizing the criterion, we take the opposite sign for the velocity.

General results for shape derivative of divergences within the exponential family

In this paper, we consider that pdfs belong to the exponential family. In this case, the current pdf estimate q(y, Ω) is now indexed by a set of parameters θ ∈ Θ ⊂ R κ (e.g. we have κ = 2 and θ = (µ, σ) T where µ is the mean and σ the variance for the Normal family). When using the exponential family, we rather index the pdf by η which is the natural parameter as explained below. In order to derive the criterion, we must take into account the dependence of the natural parameter with the domain. We then restrict our study to the full rank κ-parameter canonical exponential family [START_REF] Bickel | Mathematical statistics: basic ideas and selected topics[END_REF]. For this family, we can establish a 1-1 correspondence between η and Ω and so compute directly the shape derivative of D(Ω).

In the sequel, let us first introduce the exponential family and some properties and then explain the computation of the shape derivative. We then specialize our result when parameters are estimated using the Maximum Likelihood Estimation (MLE) method. We also give some results for the optimization of the Kullback-Leibler (KL) divergence. In this case, the shape derivative reduces to a very simple general expression.

The exponential family: Definition and properties

The multi-parameter exponential family [START_REF] Koopman | On distributions admitting a sufficient statistic[END_REF] is naturally indexed by a κ-dimensional real parameter vector η and a κ-dimensional natural statistic vector T(Y ). We draw the reader's attention to the fact that η is a function of θ ∈ Θ which is the parameter of interest in most applications (for the Gaussian distribution, we have θ = (µ, σ) T ).

Definition 1. The family of distributions of a Random Variable (RV) Y {q

θ : θ ∈ Θ ⊆ R κ },
is said a κ-parameter canonical exponential family, if there exists real-valued functions:

• η(θ) = [η 1 , ..., η κ ] T with η i : Θ ⊆ R κ → R • h : R → R • B : Θ → R • T = [T 1 , ..., T κ ] T : R κ → R
such that the pdf q θ (y) may be written as:

q θ (y) = h(y) exp[ η(θ), T(y) -B(θ)] with y ∈ χ ⊂ R . ( 5
)
The term T is called the natural sufficient statistic and η the natural parameter vector. The term η, T denotes the scalar product.

Letting the model be indexed by the natural parameter η rather that θ, the canonical κ-parameter exponential family generated by T and h is defined as follows:

q η (y) = h(y) exp[ η(θ), T(y) -A(η)] , (6)
with

A(η) = log R +∞ -∞ h(y) exp[ η(θ), T(y) ]dy. The natural parameter space is defined as E = {η ∈ R κ ; -∞ < A(η) < +∞}.

Some common distributions

Table 1 provides a synthetic description of some common distributions of the exponential family:

Distribution θ T η(θ) T T(y) T A(η) Normal (µ, σ 2 ) ( µ σ 2 , -1 2σ 2 ) (y, y 2 ) 1 2 - η 2 1 2η 2 -log -η 2 π Gamma (λ, p) (-λ, p -1) (y, log y) -(η 2 + 1) log -η 1 + log Γ(η 2 + 1) Beta (r, s) (r -1, s -1) (log y, log(1 -y)) -log B(η 1 + 1, η 2 + 1) Poisson µ log µ y e η Exponential λ -λ y -log -η Rayleigh θ 2 -1/2θ

Properties

The following results will be useful for our RBAC scheme based on the exponential family. Their proofs may be found in [START_REF] Bickel | Mathematical statistics: basic ideas and selected topics[END_REF]. These properties give us a relation between the parameters η and the domain Ω through the use of the expectation of the natural statistics T(Y ). The first theorem provides general relations between the gradient of A and the expectation of T(Y ) while the second theorem allows us to establish a 1 -1 correspondence between η and E[T(Y )] (for the full rank exponential family). Such a relation may then be used to express the parameter η and derive it according to the domain.

1. E is convex. 2. A : E → S ⊆ R is convex. 3. E[T(Y )] = ∇A(η). 4. Cov[T (Y )] = Ä(η).
where ∇A = ( ∂A The following theorem establishes the conditions of strict convexity of A, and then those for ∇A to be 1-1 on E. This is a very useful result for optimization (derivation) purposes: space E, we have [START_REF] Bickel | Mathematical statistics: basic ideas and selected topics[END_REF]:

1. η → ∇A(η) is 1-1 on E.

The family may be uniquely parameterized by µ(η) ≡ E[T(Y )] = ∇A(η).

The anti-log-likelihood function is a strictly convex function of η on E.

These results establish a 1-1 correspondence between η and E[T(Y )] such that:

µ = ∇A(η) = E[(T(Y )] ⇔ E η = φ (E[T(Y )]) , ( 7
)
holds uniquely with ∇A and φ continuous.

Estimation of the hyperparameters The relation 7 allows us to express the parameter η as a function of E[(T(Y )]. In order to estimate the parameters, we replace E[T(Y )]

by the empirical estimate of the mean T(Y ). This corresponds to the MLE of the parameter. Indeed, the MLE of η corresponds to minimizing the anti-log-likelihood score (for independent and identically distributed (iid) data). By differentiation of the antilog-likelihood according to η, we find ∇A(η MLE ) = T (Y ). Note however that in this case, this is the discrete sample mean. The following example illustrates this stating:

Example 1. When dealing with the Rayleigh distribution, we have η = -1 2θ 2 , A(η) = log(-2η) and T (y) = y 2 . By computing A (η) = T(Y ), we find that -

1 η = 1 |Ω| R Ω y(x) 2 dx,
which corresponds to the MLE of the parameter θ 2 given by θ

2 ML = 1 2|Ω| R Ω y(x) 2 dx.

Shape derivative of the criterion

In this section, we propose to derive according to the domain the functional [START_REF] Kass | Snakes : Active contour models[END_REF]. The dependence of the functional with the domain is due to the estimation of the parameter η detailed above. In the sequel, for the sake of simplicity, we will invariably denote η for the natural parameter and its finite sample estimate over the domain (without a slight abuse of notation, this should be η). We are now ready to state our main result:

Theorem 3. The G âteaux derivative, in the direction of V, of the functional (1), is:

< D (Ω), V >= ∇ V η, C , ( 8
)
where

∇ V η = [< ∇η 1 (Ω), V >, ..., < ∇η κ (Ω), V >]
is the Gâteaux derivative of η in the direction of V, ., . is the usual scalar product of two vectors and:

C = E[∂ 1 Ψ(q(Y, η(Ω)), p(Y ))(T(Y ) -E[T(Y )])].
The term ∂ 1 Ψ denotes the partial derivative of Ψ according to the first variable.

The proof is detailed in Appendix 5.2.

We then have to compute the shape derivative ∇ V η. Such a computation requires an estimation of the expectation E[T(Y )] as explained in the next section.

Computing the shape derivative for the MLE estimator

As mentioned in section (3.1.3), the expectation E[T(Y )] can be replaced with the empirical estimate of the mean T(Y ) which is computed over the considered domain Ω. Using such an estimation for the hyperparameter, we can state the following proposition : Lemma 1. Within the full rank exponential family, and using the MLE estimator for the hyperparameters, the shape derivative ∇ V η can be expressed as:

∇ V η = Ä(η) -1 ∇ V (T) . ( 9
)
where Ä(η) -1 = I(η) -1 is the inverse of the Hessian matrix of Ä which is also the fisher information matrix I. The derivative ∇ V (T) is given by:

∇ V (T) = 1 |Ω| Z ∂Ω T(y) -T(y(a)) (V • N)da(x) , ( 10
)
The proof is given in Appendix 5.3. We can then replace the shape derivative of the natural parameters given in Lemma 1 in the general Theorem 3. The corollary that gives the shape derivative then follows:

Corollary 1. The G âteaux derivative, in the direction of V, of the functional (1), is:

< D (Ω), V >= 1 |Ω| Z ∂Ω κ ∑ i=1 C i κ ∑ j=1 [ Ä(η)] -1 i j (T j (y) -T j (y(a))) (V • N)da ,
where the κ components of the vector C are defined as follows:

C i = E[∂ 1 Ψ(q(Y, η(Ω)), p(Y ))(T i (Y ) -T i (Y )] i ∈ [1, κ].
The term ∂ 1 Ψ denotes the partial derivative of Ψ according to the first variable.

In order to fix ideas, the functional D(Ω) can be chosen as the Kullback-Leibler divergence, in this case ∂ 1 Ψ(q, p) = logq+ 1-log p-p q . In order to compute the vector C in Corollary 1, we can assume that the pdf p belongs to the exponential family and to the same parametric law as the pdf q. Let us denote by η 1 the parameter of the pdf p. This parameter is supposed to be already computed or dependent of another domain and so does not depend on the domain Ω. We then state the following proposition: Lemma 2. When p(y, η 1 ) and q(y, η(Ω)) are two members of the exponential family that belong to the same parametric law with respective parameters η 1 and η, and when the functional D(Ω) is chosen as the KL divergence, we find for the vector C defined in Theorem 1:

C = Ä(η)(η -η 1 ) + ∇A(η) -∇A(η 1 ) .
A proof is given in appendix C. This expression demonstrates that the derivative can be very simply computed using the natural parameters and the sufficient statistics of the law. Let us give two examples of computation for both the Rayleigh and the Gaussian law.

Example 2. When dealing with the Rayleigh distribution, following example 2, with

θ 2 = 1 2 y 2 , the term C is equal to C = 2θ 2 θ 2 θ 2 1 - θ 2 1 θ 2 .
we then find for the derivative of KL divergence:

< KL (Ω), V >= 1 |Ω| Z ∂Ω C 2θ 2 (1 - y(a) 2 2θ 2 )(V • N)da(x) . ( 11
)
Example 3. When dealing with the Gaussian distribution, the term C is equal to

C = ⎡ ⎣ ( σ 2 σ 2 r + 1)(µ -µ 1 ) µ 2 + σ 2 -µ 2 1 -σ 2 1 + 2 σ 2 σ 2 1 µ 2 -µµ 1 + σ 4 ( 1 σ 2 1 -1 σ 2 ) . ⎤ ⎦ ( 12
)
We then find for the derivative of KL divergence:

< KL (Ω), V >= 1 σ 2 |Ω| Z ∂Ω -(y -µ) C 1 (1 + 2µ σ 2 ) -C 2 µ σ 2 + (y 2 -σ 2 -µ 2 ) C 1 µ σ 2 - C 2 2σ 2 (V • N)da(x) .

Maximisation of divergences

In this section, we propose to concentrate on the segmentation of an image into two regions (namely Ω and its complement Ω c ) by maximizing the criterion 2.

Evolution equation

When using the MLE estimator for the parameters, and noting that Ω and Ω c shares the same boundary with opposite normals, we take

T(y) = 1 |Ω| R Ω T(y(x))dx and T(y) c = 1 |Ω c | R Ω c T(y(x))dx. Using Corollary 1 and the fact that < D (Ω, Ω c ), V >= ∇ V η, C + ∇ V η c , C c ,
we find for the evolution equation:

∂Γ ∂τ = 1 |Ω| κ ∑ i=1 C i (Ω) κ ∑ j=1 Ä(η) -1 i j (T j (y) -T j (y(x)) - 1 |Ω c | κ ∑ i=1 C i (Ω c ) κ ∑ j=1 Ä(η c ) -1 i j (T j (y) c -T j (y(x)) N.
For the KL divergence, the term C is evaluated as explained in section 3.3. A classical regularization term λκ is added where λ is a positive constant and κ the curvature. As far as the numerical implementation is concerned, we use the level set method approach first proposed by Osher and Sethian [START_REF] Osher | Fronts propagating with curvature-dependent speed: Algorithms based on hamilton-jacobi formulation[END_REF].

Comparison with other methods in the Gaussian case

In this section, we propose to compare the behavior of our data term based on the maximization of the symmetrized Kullback-Leibler divergence between parametric pdfs to two other well-known region-based methods [START_REF] Zhu | Region competition: unifying snakes, region growing, and bayes/MDL for multiband image segmentation[END_REF][START_REF] Chan | Active contour without edges[END_REF]. The first method is the famous Chan & Vese method [START_REF] Chan | Active contour without edges[END_REF]. Such a criterion implies a Gaussian distribution for the feature y with a fixed variance. The corresponding evolution equation can be found in [START_REF] Chan | Active contour without edges[END_REF].

The second method has been first proposed by [START_REF] Zhu | Region competition: unifying snakes, region growing, and bayes/MDL for multiband image segmentation[END_REF] and aims at minimizing the anti-loglikelihood for a Gaussian distribution. The evolution equation can be found in [START_REF] Zhu | Region competition: unifying snakes, region growing, and bayes/MDL for multiband image segmentation[END_REF].

In order to compare these terms, let us express the non symmetrized KL divergence using the expectation under the pdf q, denoted by E q , as follows:

D(q p) = E q [log(q(Y, η Ω ))] -E q [log(p(Y, η Ω c ))] (13)
To get the gist of using KLD as a criterion in an RBAC functional, consider the data y i = {y(x)|x ∈ Ω} as an iid sequence from the statistical model q(y, η Ω ). Using the weak law of large number for a very large domain Ω, the first term (which corresponds to the entropy) can then be expressed as 1

|Ω|

R

Ω log(q(y(x), η(Ω))dx. Maximizing the first term in KL divergence can then be seen as equivalent to minimizing the anti-log-likelihood score [START_REF] Cover | Elements of Information Theory[END_REF] divided by the size of the sample (which corresponds to the entropy under the law of large number). Using the same assumptions, the second term of KL divergence can be seen as the minimization of the plausibility of the data provided by Ω c in the inside region Ω. When using the symmetrized version, we act both on Ω and Ω c .

Let us now compare experimentally the behavior of these criterions for the extraction of an homogeneous region corrupted by a Gaussian noise in an image. We propose to take the example of the segmentation of the White Matter (WM) in T1-weighted brain MRI images. We perform the three evolution equations using the Gaussian assumption for the pdf of the feature y within each region. The feature y is chosen as the Intensity of the image. The initial contour is given in Figure 1.(a) and we also show the two initial pdfs (b), namely q η (I, Ω) which corresponds to the distribution of the intensity I inside the region Ω and q η c (I, Ω c ) which corresponds to the distribution of I inside the region Ω c (i.e. outside the region ). In Figure 1, we can observe the final active contour obtained using our criterion (22) and the two other criterions mentioned above. We can remark that our criterion acts as an extractor of the most important Gaussian in the initial mixture of Gaussian (see Figure 1.e). The two other criterions separate the mixture without extracting a single Gaussian. So, with our method, we can directly obtain the White Matter of the brain without a multiphase scheme.

Examples of applications

In this part, we consider two examples of application (brain MRI images and contrast echocardiogaphy) using two different noise models (Gaussian and Rayleigh).

Concerning 3D T1-weighted MRI images of the brain, the noise model is assumed to be represented by a Rician distribution [START_REF] Goodman | Some fundamental properties of speckle[END_REF]. For large signal intensities the noise distribution can be considered as a Gaussian distribution (this is the case for the White Matter (WM) or the Gray Matter (GM)). We propose in Figure 2 an example of WM segmentation by maximizing the KL divergence between Gaussian distributions. When evaluating quantitatively our results of WM segmentation on the simulated brain T1weighted MRI images provided by the Montreal Neurological Institute Brain Web URL, we find a dice coefficient of 0.91, a very law False Positive Fraction (FPF) of 0.8% and a True Positive Fraction (TPF) of 84%. As the Rayleigh distribution is well suited to model noise in echography [START_REF] Goodman | Some fundamental properties of speckle[END_REF], this noise model was applied for segmentation of the left ventricle in contrast echocardiography. Final contours for several images of the sequence are shown in Figure 3. The segmentation is accurate all along the sequence. Note that experimental results reported in [START_REF] Lecellier | Statistical region-based active contours with exponential family observations[END_REF][START_REF] Martin | Influence of the noise model on level set active contour segmentation[END_REF] prove that when using the appropriate noise model, segmentation results are more accurate and less sensitive to the choice of the regularization parameters.

Shape derivation tools

Let us remind this useful theorem [START_REF] Delfour | Shape and geometries[END_REF] that will be used in the following proofs.

Theorem 4. The Gâteaux derivative of the functional J(

Ω) = R Ω f (x, Ω) dx in the direction of V is: < J (Ω), V >= R Ω f s (x, Ω, V)dx - R ∂Ω f (x, Ω)(V • N)da(x)
where N is the unit inward normal to ∂Ω, da its area element and f s the shape derivative of f [START_REF] Delfour | Shape and geometries[END_REF].

Proof of Theorem 3

To compute < D (Ω), V >, we must first get the derivative of q(y(x), η) with respect to the domain, and apply the chain rule to Ψ(q(y(x), η), p(y)). To simplify the notation we write the Eulerian derivative of η as < η (Ω), V >= ∇ V η = < η 1 (Ω), V >, .., < η κ (Ω), V > T .

Using the definition of q(y, η) given in [START_REF] Aubert | Image segmentation using active contours: Calculus of variations or shape gradients ?[END_REF] and the chain rule applied to A (η(Ω)), we obtain:

< q (y, η), V > = h(y)( ∇ V η, T(y) -∇ V η, ∇A(η) ) e η(Ω),T(y) -A(η(Ω)) (14) = q(y, η) ∇ V η, T(y) -∇A(η) .

By the chain rule applied to Ψ(q(y(x), η), p(y)), we get < Ψ (q(y, η), p(y)), V >=< q (y, η), V > ∂ 1 Ψ(q, p), which gives < D (Ω), V >= R χ q(y, η)∂ 1 Ψ(q, p) ∇ V η, T(y) -∇A(η) dy. We introduce C = R χ q(y, η)∂ 1 Ψ(q, p)(T(y) -∇A(η)) dy = E[∂ 1 Ψ(q, p)(T(Y ) -E[T(Y )])] which completes the proof.

Proof of Lemma1

When using the MLE, the term E[T(Y )] can be empirically estimated with T(Y ) and so derived easily with respect to the domain Ω. We propose to directly derive the expression ∇A(η) = T(Y ) which gives:

κ ∑ j=1 < η j , V > ∂ 2 A ∂η i ∂η j (η) =< T i (Y ) , V > ∀i ∈ [1, κ] , (15)
which can be written in the compact form ∇ V (T) = Ä(η)∇ V η.

Restricting our study to the full rank exponential family, where Ä(η) is a symmetric positivedefinite, hence invertible, matrix (Theorem 2), the domain derivative of the parameters η is uniquely determined by Ä(η) -1 ∇ V (T) = ∇ V η where ∇ V (T) is given by: ∇ V (T) = 1 |Ω| R ∂Ω T(y) -T(y(a)) (V • N)da(x) (taking benefit of theorem 4) and the lemma follows.

Proof of Lemma 2

Since p and q belongs to the same parametric law, they share the same value for h(y), T(y) and A(η) and then log(q)log(p) = η -η 1 , T(y) -A(η) + A(η 1 ). The value of C is then C = s 1s 2 , with:

s 1 = E[( η -η 1 , T(y) -A(η) + A(η 1 ) + 1)(T i (Y ) -E[T i (Y )]] s 2 = E[ p q (T i (Y ) -E[T i (Y )]]E p [(T i (Y ) -E[T i (Y )]]
Developing the expression of the expectation of the second term,we find s 2 = E p [(T i (Y )-E[T i (Y )]] = ∇A(η 1 ) -∇A(η). Using the linearity of the expectation and the fact that

E[T j (Y )(T i (Y )] -E[T i (Y )]E[T j (Y )
] designates the covariance matrix of the sufficient statistics T and can then be replaced by Ä(η) i j = Cov[T(Y )] i j = Ä(η) ji , we find: s 1 = ∑ κ j=1 (η j -η 1 j ) Ä(η) i j , and then C = Ä(η)(η -η 1 ) + ∇A(η) -∇A(η 1 ).

Theorem 1 .

 1 Let {q η : η ∈ E} a κ-parameter canonical exponential family with natural sufficient statistic T(Y ) and open natural parameter space E, we then have the following properties:

)

  T represents the gradient of A, and Ä is the Hessian matrix of Awith Äi j = ∂ 2 A ∂η i ∂η j .

Theorem 2 .

 2 Let {q η : η ∈ E} a full rank (i.e. Cov[T (Y )] is a positive-definite matrix) κ-parameter canonical exponential family with natural sufficient statistic T(Y ) and open natural parameter

Fig. 1 .

 1 Fig. 1. T1-weighted brain MRI segmentation results (extraction of the White Matter). The pdf of the intensity inside the contour is in solid line, the pdf of the intensity outside the contour is in dotted lines. (a): initial contour and (b) : associated pdfs, column (c): final contour and pdfs for the Chan & Vese method[START_REF] Chan | Active contour without edges[END_REF], column (d): for the log-likelihood method[START_REF] Zhu | Region competition: unifying snakes, region growing, and bayes/MDL for multiband image segmentation[END_REF], column (e): for the maximization of the KL divergence.

Fig. 2 .

 2 Fig. 2. 3D Segmentation of WM in a T1 brain MRI using KL maximization

Fig. 3 .

 3 Fig. 3. Segmentation of the LV in a contrast echocardiographic sequence

Table 1 .

 1 Some common canonical exponential families. B(α, β) is the Euler Beta function.

	2	y 2	-log -2η