Matthieu Kowalski
email: kowalski@cmi.univ-mrs.fr

Bruno Torrésani
email: bruno.torresani@cmi.univ-mrs.fr

Sparsity and persistence: mixed norms provide simple signal models with dependent coefficients

Keywords: Mixed-norms, Time-frequency decompositions, Sparse representations

Sparse regression often uses p norm priors (with p < 2). This paper demonstrates that the introduction of mixed-norms in such contexts allows one to go one step beyond in signal models, and promote some different, structured, forms of sparsity. It is shown that the particular case of the 1,2 and 2,1 norms leads to new group shrinkage operators. Mixed norm priors are shown to be particularly efficient in a generalized basis pursuit denoising approach, and are also used in a context of morphological component analysis. A suitable version of the Block Coordinate Relaxation algorithm is derived for the latter. The group-shrinkage operators are then modified to overcome some limitations of the mixed-norms. The proposed group shrinkage operators are tested on simulated signals in specific situations, to illustrate and compare their different behaviors. Results on real data are also used to illustrate the relevance of the approach.

Introduction

Sparse approximation approaches have enjoyed considerable popularity in recent signal processing applications. Sparsity seems to be a particularly efficient guiding principle in view of a number of tasks such as signal compression, denoising, image de-blurring, blind source separation,. . . The guiding principle may be summarized as follows: for most signal classes, it is possible to find a basis or a dictionary of elementary building blocks (or atoms) with respect to which all (or most) signals in the class may be expanded, so that when the expansion is truncated in a suitable way, high precision approximations are obtained even when very few terms are retained. A large number of signal and image processing "success stories" may be described in such a way, including image compression and denoising using wavelets, curvelets, or more sophisticated *-lets, audio coding using MDCT bases, and so forth. Several efficient sparse expansion algorithms have been proposed, including among others simple expansion with respect to a fixed basis followed by soft or hard coefficient thresholding, iterative thresholding strategies in redundant dictionaries, greedy (pursuit) algorithms, or more elaborate approaches such as sparse regression in Bayesian frameworks. Thresholding and iterative thresholding strategies are particularly interesting, mainly because thresholding automatically generates sparsity. In addition, corresponding algorithms are easy to implement and generally exhibit fast convergence properties.

A main strength of these thresholding approaches is that they process the signal representation coefficientwise, which results in low complexity algorithms. However, this may become a weakness when it comes to applications to real signals. Indeed, the assumption of coefficient independence is generally not realistic. For example, when using wavelet or local cosine bases for expanding 1D signals, abrupt changes manifest themselves by groups of time-localized large coefficients, and frequency modulated signals exhibit ridges of frequency localized large coefficients. The same remark applies to edges and regular textures in wavelet or local cosine representations of images. Several different approaches have been considered to handle such dependencies between coefficients, including structured versions of matching pursuit (for example, harmonic or molecular versions of matching pursuit), coefficient domain modelling, or construction of suitable bases. Here, we propose to keep the coefficient modelling approach. However, rather than introducing explicit models for coefficients, we follow the thresholding and iterative thresholding approaches and design new group thresholding methods, associated with mixed norms in the coefficient domain.

More precisely, we consider the following problem. Let y ∈ R T be a noisy observation of a signal s ∈ R T . Let D denote a fixed dictionary for R T , and denote by A ∈ R T ×N the matrix whose columns are the vectors from the dictionary D. We assume that s has a sparse expansion in D, and we want to estimate s from y. A classical estimate is given by the basis pursuit denoising approach introduced by Donoho and coworkers [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF], also known as Tibshirani's LASSO estimate [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]. The estimate is obtained by the following optimization:

x = argmin x∈R N y -Ax 2 2 + λ x 1 (1)
where λ ∈ R is a fixed parameter, so that, Ax is the estimate of s. The 1 norm directly leads to soft thresholding strategies. Similar algorithms may be derived using more general p norms, i.e. replacing x 1 with x p p . That estimate treats all coefficients independently. Dependencies between selected subsets of coefficients may be introduced as soon as the latter may be labelled using a double index (for example, a time-frequency index), say x = {x ab , a = 1, . . . N a , b = 1, . . . N b }. Then a new estimate is obtained by replacing the 1 norm in (1) with a mixed norm, namely by solving for

x = argmin x∈R N y -Ax 2 2 + λ   Na a=1 N b b=1 |x a,b | p q/p   1/q . ( 2
)
Here, the roles of indices a and b are purely conventional. However, permuting a and b corresponds to a different problem. The . p,1 mixed norms have been used by various authors to model a "joint sparsity" of coefficients in the context of multichannel signals, using the FOCUSS algorithm [START_REF] Cotter | Sparse solutions to linear inverse problems with multiple measurement vectors[END_REF], greedy pursuits [START_REF] Tropp | Algorithms for simultaneous sparse approximation. part I: Greedy pursuit[END_REF], convex relaxation [START_REF] Tropp | Algorithms for simultaneous sparse approximation. part II: Convex relaxation[END_REF] or iterative thresholding strategies [START_REF] Fornasier | Recovery algorithm for vectorvalued data with joint sparsity constraints[END_REF][START_REF] Teschke | An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector valued regimes and an application to color image inpainting[END_REF].

It is worth noticing that like the LASSO method and p generalizations, the mixed norm approach admits a simple Bayesian interpretation, assuming Gaussian white noise (which justifies the choice of the 2 norm for the data fidelity term), and a coefficient prior of the form

f (x) ∝ exp -λ x q p,q
, which explicitly introduces couplings between coefficients.

This prior still assumes independence between groups of coefficients. We show that this independence assumption may be relaxed by modifying the design of the group-shrinkage operators used to solve (2) when A is orthogonal.

Mixed norms can also be implemented into multilayered type signal expansions, such as the ones used in [START_REF] Berger | Removing noise from music using local trigonometric bases and wavelet packets[END_REF][START_REF] Daudet | Hybrid representations for audiophonic signal encoding[END_REF][START_REF] Daudet | Towards a hybrid audio coder[END_REF] for audio signals, or in the Morphological Component Analysis (MCA for short) for images [START_REF] Starck | Image decomposition via the combination of sparse representations and a variational approach[END_REF][START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF]. The goal of MCA is to minimize functionals of the type

Φ(x 1 , x 2 ) = x 1 1 + x 2 1 + λ y -A 1 x 1 -A 2 x 2 2 2 (3)
where A 1 and A 2 are the matrices corresponding to two dictionaries, chosen to be able to sparsely describe edges and textures respectively. A similar approach may be followed to separate transient and tonal layers in audio signals. According to the discussion above, we shall show that the two 1 norms in the latter expression can be conveniently replaced with suitable mixed norms, to enforce relevant dependencies between coefficients.

The paper is organized as follows. Section 2 recalls the definition of mixed norms and introduces new groupshrinkage operators associated with these norms. This section also presents the mixed norm-based multilayered expansion on union of bases. In Section 3, the shrinkage operators are modified in order to overcome some limitations of the mixed norms. All these operators are used on simulated signals to illustrate their behavior. Some applications are presented in Section 4.

Mixed norms and thresholding

We give in this section the definition of the mixed norms we shall be interested in. For the sake of simplicity, we shall stick to the case of two indices, even though extensions are clearly possible.

Mixed norms

We are concerned with doubly labelled sequences x a,b , a = 1, . . . N a , b = 1, . . . N b . Let us start by introducing the mixed norms.

Definition 1 Let x ∈ R N , labelled by a double index (a, b) . Let p ≥ 1 and q ≥ 1, then one can define two mixed norms 1;p,q and 2;p,q on x

x 1;p,q =   Na a=1 N b b=1 |x a,b | p q/p   1/q , ( 4
)
x 2;p,q =   N b b=1 Na a=1 |x a,b | p q/p   1/q . ( 5
)
The cases p = +∞ and q = +∞ are obtained by replacing the corresponding norm by the supremum.

Mixed norms have been used extensively by mathematicians in functional analysis (see for example [START_REF] Samarah | A Shur test for weighted mixed-norm spaces[END_REF] and references therein). Here, we limit ourselves to the finite dimensional case, and focus on the particular cases •;1,2 and •;2,1 . For the sake of simplicity, we will use the 1;p,q norm for the theoretical study, and then denote it simply by p,q . The second case is obtained by simply switching the roles of a and b. In the numerical applications described in section 4 the choices will be specified precisely.

The reader may think of these two indices as the indices of a time-frequency signal expansion. However, let us stress that the developments below are not specific at all to time-frequency signal representations, and apply to any situation where signals are expanded with respect to a dictionary with two indices. Another simple example of that is multichannel signals, where a first index labels (scalar) dictionary elements and a second one labels channels. In an even more general situation, any discrete signal expansion may be re-labelled so as to be processed by our approach.

The two indices shall be used in hierarchical way: coefficients are split into independent groups, and coefficients within the same group are dependent. In this work, we will highlight this hierarchy by denoting the indices by g (for group) and m (for member) respectively. Using these notations, we shall label vectors x ∈ R N such that the p,q mixed norm of x reads

x p,q =   G g=1 M m=1 |x g,m | p q/p   1/q
, with G the number of groups and M the number of members in each group, so that N = G × M .

Remark 1 Actually, nothing forces the number of members to be the same for all groups. However, by adding "phantom" members equal to zero, one can artificially come back to the simplest situation where all groups have the same size. For the sake of simplicity, we only consider that simple case here.

It is interesting to stress that a p,q mixed norm can be seen as a "composition" of p and q norms, and therefore inherits of their properties (in particular convexity for p, q ≥ 1). With the above notations,

x p,q = G g=1 x g q p 1/q = ( x 1 p , . . . , x G p ) q . ( 6
)
For p < 2, p norms are often used as diversity measures, and minimizing the p norm of a coefficient sequence of a signal generally aims at promoting concentration for the expansion: the distribution of coefficients is more sharply peaked at the origin for p < 2. For p ≤ 1, concentration becomes sparsity, since small coefficients are forced to zero. The case p = 1 has a particular status, since the 1 norm promotes sparsity and remains convex. The situation with mixed norms is a bit more tricky, since two exponents have to be taken into account. However, we shall see below that values of p (or q) smaller than 2 still yield some form of concentration, in a somewhat structured way. More precisely, depending on the choice of p and q, concentration is promoted on each individual variable x g,m if p is close to 1, and on an entire group of variables if q is close to 1.

Group-shrinkage operators

We first introduce generalized shrinkage operators, extending LASSO and Group-LASSO (G-LASSO) estimators, before turning to extensions to the multilayered case. For the sake of simplicity, we shall concentrate on the particular values (p, q) ∈ {1, 2}.

Given an observation y ∈ R N , let us denote, for

x ∈ R N , Φ p,q [x] = 1 2 y -Ax 2 2 + λ q x q p,q , (7)
We want to solve the following optimisation problem

P p,q : x = argmin x∈R N Φ p,q [x]
in the particular case where A is an orthogonal matrix.

For the sake of simplicity, let us introduce the following notation

ȳ = A T y . (8)
Then problem P pq can also be written min

x∈R N   1 2 ȳ -x 2 2 + λ q G g=1 M m=1 |x g,m | p q/p   , (9)
Focusing on the case p, q ∈ {1, 2}, and denoting by g and m the indices as explained before (i.e. x = {x g,m }), we focus on problems P 1,2 and P 2,1 .

The solution is given in Proposition 1 below. Let us first introduce some notations. For x ∈ R, we shall set (y g,m -yg,Mg(λ) ) , and set

x + = x if x ≥ 0 and x + = 0 if x ≤ 0. For τ ∈ R + ,
|ȳ g | = Mg(λ) m=1 yg,m = yg,1:Mg(λ) 1 ,
where yg,1:Mg(λ) denotes the vector {y g,1 , . . . yg,Mg(λ) }.

Proposition 1 Let A be an orthogonal matrix.

(a) The solution x of problem P 1,2 is given by the following shrinkage operation: for all g, m

xg,m = S τg (ȳ g,m ) ,
where the group dependent threshold τ g reads

τ g = λ 1 + λM g (λ) |ȳ g | (b)
The solution x of problem P 2,1 is given by the following shrinkage operation: for all g, m

xg,m = ȳg,m 1 - λ ȳg 2 + .
Remark 2 The solution of P 2,1 is known in the statistical community as the G-LASSO estimate, and the result was given in [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF]. The solution of problem P 1,2 is obtained in [START_REF] Kowalski | Sparse regression using mixed norms[END_REF] as a part of a more general result. In contrast with the G-LASSO, we call the problem P 1,2 the Elitist-LASSO (E-LASSO, see below). Notice that in both cases, the result is a generalized soft thresholding, or shrinkage, that is applied to a group of coefficients rather than single coefficients. Hence, coefficients are not processed independently any more.

It is important to stress the striking difference between the two new shrinkage operators. In the second case (the G-LASSO case), a 1D group of coefficients is either globally retained or discarded. This may be understood as an united group shrinkage, since the same threshold applies to all members of a given group. In the first case, each coefficient is shrunk individually, but the corresponding threshold depends on its 1D neighborhood. That one can be understood as an elitist group shrinkage, since most members of a given group are thresholded, and only the emerging coefficients of each group (it may be shown that at least one coefficient is kept, see [START_REF] Kowalski | Sparse regression using mixed norms[END_REF]) remain. The difference between these two situations will appear clearly in the numerical results below. There, we also present an approximate solution of P 1,2 , which turns out to be computationally simpler.

It is also interesting to remark that the solution of P 1,2 , which only involves soft thresholdings with variable threshold values, is also the solution of the problem min

x∈R N G g=1 1 τ g M m=1 |y g,m -[Ax] g,m | 2 + λ x 1 , (10)
i.e. a sparse regression problem, with 1 sparsity prior, and in which the data fidelity term involves a data dependent weighting. From a Bayesian point of view, such a re-interpretation shows that beyond the Gaussian white noise case, the so obtained solution may also be expected to perform well in situations where a sparse signal is embedded into a noise whose variance varies as a function of g in the coefficient domain.

Multilayered expansion on union of bases

After having found the solution of problem P pq in the simple case where A is an orthogonal matrix (corresponding to an orthonormal basis), we now address similar problems, in which A is a concatenation of orthogonal matrices (corresponding to an union of orthonormal bases), and the coefficient priors are different for each basis. A motivation for this problem is the decomposition of audio signals into three layers Transient + Tonal + Noise, using MDCT bases with different time-frequency resolutions. Similar problems may also be found in image processing, under the name of Cartoon + Texture + Noise image decompositions. Such problems have been studied by various authors and a few algorithms are already available. Probability-based approaches have been used in the audio domain (see [START_REF] Molla | An hybrid audio scheme using hidden Markov models of waveforms[END_REF][START_REF] Févotte | Sparse linear regression with structured priors and application to denoising of musical audio[END_REF], that exploit simultaneously sparsity and persistence. Variational approaches (such as the so-called Morphological Component Analysis) were generally preferred in the image processing literature, that did not so far integrate the notion of persistence. The mixed norm approach we focus on represents a good compromise between the other two approaches, as it allows one to incorporate persistence in variational formulations.

We start from an optimization problem similar to the one given by MCA, but, instead of using two 1 norms to estimate the tonal and transient layers, we will use suitable mixed-norms. So that, we will minimize the following functional

Φ(x, x) = y -A(x, x) T 2 2 + λ x q p,q + µ x q p,q (11)
where the p,q and p,q norms will be chosen adequately.

To decompose a signal into several layers, one chooses a suitable dictionary for each layer. In the audio signal example, the transient layer is known to be sparsely represented in dictionaries of wavelets, or time-frequency dictionaries (like Gabor or MDCT) with a narrow window. At the opposite, the tonal layer is known to be sparsely represented in time-frequency dictionaries with a wide window.

Here we choose the special case where each dictionary is an orthonormal basis, for example, two MDCT bases with two different sizes for the windows, and apply the Block Coordinate Relaxation method [START_REF] Bruce | Block coordinate relaxation methods for nonparametric signal denoising[END_REF] (BCR for short) which inspired the Morphological Component Analysis (MCA) algorithms [START_REF] Starck | Image decomposition via the combination of sparse representations and a variational approach[END_REF]. BCR is specially adapted to unions of orthogonal bases, and is known to converge to a minimum of the basis-pursuit denoising objective functional [START_REF] Bobin | Morphological component analysis for sparse multichannel data: Application to inpainting[END_REF].

Let us introduce some notations. We denote by U and V the two bases under consideration, and by U and V the corresponding matrices. We denote by x U the coefficients corresponding to the basis U and x V the coefficients corresponding to the basis V. So that, U x U corresponds to the tonal layer and V x V to the transient layer. To obtain estimates for the two layers, we then choose to minimize the following functional

Φ(x U , x V ) = 1 2 y-U x U -V x V 2 2 + λ q x U q p,q + µ q x V q p,q (12)
The BCR algorithm is then slightly modified in order to yield a minimizer of ( 12):

Algorithm 1 -Let x (0) U ∈ R N and x (0) V ∈ R N -Do 1. r (i) U = y -V x (i) V 2. Find an estimate x (i+1) U by solving x (i+1) U = argmin x∈R N 1 2 y -U x 2 2 + λ q x q p,q
using Proposition 1 3. r

(i) V = y -U x (i+1) U 4. Find an estimate x (i+1) V by solving x (i+1) V = argmin x∈R N 1 2 y -V x 2 2 + µ q x q p,q
using Proposition 1 Until convergence Following the proof given in [START_REF] Bruce | Block coordinate relaxation methods for nonparametric signal denoising[END_REF] for the BCR algorithm, one can exploit the results of [START_REF] Tseng | Dual coordinate ascent methods for non-stricly convex minimization[END_REF] and state Theorem 1 Let U, V ∈ R N ×N be two orthogonal matrices. Let y ∈ R N and p ≥ 1, q ≥ 1 p ≥ 1, and q ≥ 1. Then Algorithm 1 converges to a minimum of [START_REF] Fornasier | Recovery algorithm for vectorvalued data with joint sparsity constraints[END_REF].

Remark 3 Here, we do not deal with the more general case where the dictionary is an arbitrary frame or even an union of frames. This case was studied in [START_REF] Kowalski | Sparse regression using mixed norms[END_REF], where convergence of a corresponding iterative thresholded Landweber algorithm was proven. However, let us point out that the approach used in the MCA algorithm could also be used if U and V are two frames. The heuristics of this algorithm is to decrease the parameters λ and µ in (3) during the iterations. Numerical results seem to indicate convergence to the global minimum, though no formal proof has been given so far (to our knowledge). This heuristics was then generalized by the Stagewise Matching Pursuit [START_REF] Donoho | Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit[END_REF].

Group shrinkage in practice: simulations

The main goal of this section is to illustrate and comment on the choice of the shrinkage operator, for specific problems on simulated signals. To this end, we limit ourselves to decompositions on an orthogonal basis (i.e. A is an orthogonal matrix). We introduced in the previous section two particular generalized shrinkage operators (G-LASSO and E-LASSO), with two completely different behaviors in an orthogonal basis. Here we analyze and illustrate the behaviors of these two approaches, and propose alternatives that overcome some potential shortcomings in specific situations.

To this end, we applied G-LASSO and E-LASSO and variants to simulated signals, specifically designed to illustrate their behavior. The simulated signals were obtained as follows. First, the time-frequency map Λ was simulated, using the Hidden Markov Model studied in [START_REF] Molla | An hybrid audio scheme using hidden Markov models of waveforms[END_REF], that generates persistence along the time axis. This map was then used to simulate a signal of the form

y[t] = ∈Λ x u [t] . ( 13
)
The index set Λ = { : x = 0} is called the significance map or time-frequency map. The corresponding (i.i.d) time-frequency coefficients x were simulated using a normal law N (0, 1). An example of so-generated significance map is displayed in Fig. 1. The map has 8.5% non zero coefficients. In the numerical examples shown below, we will consider maps with 8.5% and 1% non zero coefficients respectively, to study the behavior of the algorithms at different sparsity levels.

Selection of relevant groups

Relabelling

In some situations, all the members of a given group need not be active at the same time. When subgroups that are simultaneously active or inactive are known in advance, coefficients may be re-labelled so that the classical G-LASSO estimate may be used. A trivial example of such re-labelling is shown in Fig. 2, in the context of a multichannel signal. There, the re-labelling is simply a splitting of groups into subgroups, but more complex re-labellings can also be considered.

However, this is not the most general situation, and subgroups of active coefficients are generally neither known in advance, nor even fixed. For that reason, a "sliding window" alternative of the above described approach is desirable.

Windowing

For this purpose, let us now assume that some extra information about the coefficients is available, telling us for each coefficient of index k = (g, m) which are the other coefficients that are likely to be "active" or "inactive" simultaneously with k. This generates a neighborhood system, associating to any "group-member" index k = 1, . . . N a group N (k) of "close" indices. Now, for a given index k, it seems reasonable to use only its neighbors in N (k) to estimate its sparse expansion, exploiting persistence within N (k). Using the G-LASSO estimate in Proposition 1-(b), this suggests to compute

xg,m = ȳg,m 1 - λ ȳk 2(N (k)) + , (14)
where we have denoted by ȳk the subsequence ȳk = {ȳ k , k ∈ N (k)} .

We call this estimate the Windowed Group-LASSO (WG-LASSO). Notice that unlike the re-labelling approach alluded to above, each coefficient k uses its own neighbors, instead of the whole group. Compared to Proposition 1-(b), the estimated coefficients xg,m are obtained from the observations ȳg,m = [Ay] g,m by pointwise multiplication with a mask function, which now depends on the index k = (g, m). Notice also that this new generalized thresholding is not any more associated to a simple variational problem. Fig. 3 shows an example of a sliding window used to group a channel with its neighborhood, where m is the channel index and g the time-frequency index.

Simulations

Here we consider decomposition of multichannel signals (sampled at 44100 Hz) on a given MDCT basis (with 23.3 millisecond long window):

y m [t] = m∈Λ x g,m u g [t] , (15)
where g is a time-frequency index and m labels channels. Λ = {g : x g,m = 0} is the significance map, or time-frequency map, and is assumed to be the same for all channels. Then we denote by y = (y 1 , . . . , y M ) ∈ R N ×M the multichannel signal, organized as an N × M matrix whose columns are the channels, and x ∈ R N ×M the unknown coefficient sequences. As before, we set y = Ay ∈ R N ×M (we recall that here A is an N × N orthogonal matrix).

In this context, the groups labelled by g and the members labelled by m in the previous section correspond respectively to the time-frequency indices and the channels. In other words, the model involves "between channels" dependencies.

Two multichannel signals were simulated as follows 1. Choose a percentage of non-zero coefficients, and generate two time frequency maps Λ 1 and Λ 2 with that prescribed percentage. 2. Simulate two sets of i.i.d. N (0, 1) time-frequency coefficients x g,m , m = 1, . . . 4 and g ∈ Λ 1 (resp. m = 5, . . . 8 and g ∈ Λ 2 ). 3. Synthesize the signals using model [START_REF] Molla | An hybrid audio scheme using hidden Markov models of waveforms[END_REF].

The simulated signals have then M = 8 channels. The first four channels share time-frequency map Λ 1 and the last four share time-frequency map Λ 2 .

The various generalized thresholding estimators described above are compared in the context of a denoising problem. A Gaussian white noise is added to the multichannel signals so as to obtain a SNR equal to 10 dB. For the sake of simplicity, the SNR is not calculated channelwise, but on the the entire multichannel signal:

SN R(x, x) = 20 log 10 x 2 x -x 2 (16)
where . 2 denotes the Fröbenius norm of the multichannel signal. This SNR may differ from the mean of SNR of all the channel, but this difference is less than 1 dB and does not influence the behavior of the displayed curves.

The estimators under study are the following -LASSO, corresponding to the problem x = argmin

x∈R N ×M ȳ -x 2 2 + λ x 1 .
All "channel-time-frequency" coefficients are independent, the estimate is obtained by soft-thresholding. -G-LASSO 1, corresponding to the problem x = argmin

x∈R N ×M ȳ -x 2 2 + λ G g=1 M m=1 |x g,m | 2 .
For a given time-frequency index, all the channels are gathered to create the groups of G-LASSO. The groups are independent. This corresponds of the grouping given on Fig. 2 (left). -G-LASSO 2, which exploits prior information on the two time-frequency maps Λ 1 and Λ 2 , corresponds to min

x∈R N ×M ȳ-x 2 2 +λ G g=1   4 m1=1 |x g,m1 | 2 + 8 m2=5 |x g,m2 | 2   .
For a given time-frequency index, the first four channels are gathered into a group, and the last four into an another group. The groups are independent and correspond of the regrouping given on Fig. 2 (right). -The WG-LASSO, corresponding to the estimate given in Equation ( 14). The two nearest neighbors of a channel are gathered using a sliding window to give the estimate. This corresponds to the grouping given in Fig. 3.

Different estimates were computed for various values of λ. The range of values for λ was chosen so as to obtain estimates with different degrees of sparsity, i.e. with various numbers of coefficients set to zero: the bigger the λ, the sparser the estimate. The curves in Fig. 4 show the evolution of SNR as a function of the number of non zero coefficients (which depends on the value of λ) of the different estimates for the simulated signals using the map displayed in Fig. 1 (i.e. 8.5% nonzero coefficients). Similar results, obtained using sparser significance maps (1% nonzero coefficients) are displayed in Fig. 5.

The behavior of the estimates clearly depends on the degree of sparsity of the input signal. In the two considered cases, G-LASSO 2 (which uses more prior information than the others) reaches the best SNR, and provides an higher SNR than the other estimates when the number of selected coefficients is close to the true number of non zero coefficients.

WG-LASSO outperforms LASSO on the two curves, except for large values of the sparsity penalty (when many coefficients are set to zero); however this case must be avoided to obtain a good estimate, as the SNR collapses quickly.

Despite their globally different aspects, the curves in Fig. 4 and Fig. 5 show very similar behaviors. In both cases, best results are obtained when the groupings are known a priori. When such a prior information is unavailable, the WG-LASSO is definitely a good alternative to exploit dependences between some coefficients. In addition, if coefficients cannot be clustered into groups, but possess some neighboring relationships, WG-LASSO is able to exploit the latter.

Coefficient selection within sparse groups

Let us now turn to E-LASSO estimates. We show in this subsection some numerical results obtained using E-LASSO, a simplified version of E-LASSO, and a variant proposed in order to introduce across groups persistence.

An approximation of the E-LASSO estimate

As may be seen from Proposition 1, E-LASSO involves a sorting of coefficients ȳ, and the determination of numbers M g (λ) prior to the actual shrinkage operation. When these operations are skipped, this yields an approximation of the estimator, obtained by replacing the threshold

τ g = λ 1 + λM g (λ)
|ȳ g | by the approximation

τ g = λ 1 + λM ȳg 1 . ( 17
)
This approximation called AE-LASSO (for Approximate E-LASSO), is simpler to compute, and has a practical interpretation, in particular in the limit of large λ values. Indeed, letting λ → ∞ in Equation ( 17), the coefficient are thresholded by ȳg,m 1 /M , which is the average of the coefficients |ȳ g,m | for a fixed group index g. The main shortcoming of this approximation is that the threshold is bounded by this average value, which bounds from below the number of retained coefficients. An advantage of this approximation, is that the role of the regularization parameter is much easier to understand. We shall see in the numerical examples below that when the number of retained coefficients is large enough, AE-LASSO is actually a good approximation of E-LASSO.

Introduction of persistence

As we have seen above, the 1,2 coefficient penalty is significantly different from the 2,1 one, that leads to G-LASSO regression: it promotes sparsity within groups of coefficients instead of sparsity across groups. For example, the thresholding formula ( 17) selects a small number of coefficients within each group. To fix the ideas, let us assume that a single coefficient is retained within each group. This coefficient is likely to vary from a group to another, since nothing in the norm prevents it from doing so. If one wants to promote persistence in the retained coefficients, an approach similar to the previous one may be developed, taking into account neighbors of the considered coefficients. We start by associating to any group index g a family N (g) of neighbors. Then, for fixed g, we can solve the minimization problem with 1,2 coefficient penalty on the vector ȳN (g) = {ȳ g ,m , m = 1, . . . M, g ∈ N (g)} Applying the same approach as before, the generalized thresholding formula ( 17) is now replaced with

τ g = λ 1 + λ|N (g)| ȳN (g) 1 , (18)
|N (g)| being the cardinality of the set N (g). Again, this generalized thresholding is not associated with a simple variational approach. The corresponding estimator is termed PE-LASSO (for Persistent Elitist LASSO).

Simulations

To illustrate the behavior of the estimators described above, we simulated a signal as follows. First, a timefrequency map was generated as before (the map 1 of the previous subsection was chosen); then coefficients were generated from a normal law N (0, 1). To follow the model given by Equation ( 10), at each time index, we added a Gaussian white noise, whose variance was randomly taken from a uniform distribution (between 1 and 128). Then denoising was performed using the E-LASSO, AE-LASSO, PE-LASSO, WG-LASSO and LASSO estimates, with various values of the λ parameter. The PE-LASSO estimate was done by introducing time persistence, taking nearest neighbors into account (1 time index before and one after). WG-LASSO was performed by gathering the 4 time-neighbors of a given time-frequency coefficient. We display in Fig. 6 and 7 the SNR as a function of the number of retained non-zero coefficients for the previous estimators, for two different values of input SNR. As expected, the E-LASSO estimate performs best in this situation. The AE-LASSO estimate is close of the E-LASSO estimate, but does not allow for very small numbers of retained coefficients (as explained before). WG-LASSO performs quite well when the number of non zero coefficients is over-estimated. Finally, the PE-LASSO estimate is quite disappointing, as it only outperforms the classical LASSO. Introducing persistence into the estimator does not seem to pay, even in situations where persistence is present in the signal.

Results on real signals

We now illustrate the various approaches described above with three different problems:

-Denoising of multichannel signals, in an additive Gaussian white noise situation. -Denoising of a single channel signal, with non stationary random noise -Multilayered signal decomposition.

Multichannel denoising

Sparse approximation techniques have been extended recently to multichannel signals (see [START_REF] Bobin | Morphological component analysis for sparse multichannel data: Application to inpainting[END_REF][START_REF] Gribonval | Atoms of all channels, unite! average case analysis of multichannel sparse recovery using greedy algorithms[END_REF] and references therein). We address such a problem directly via a generalized basis pursuit denoising approach, using the 1 norm in the time-frequency direction, and the 2 norm across channels.

Let us consider a multichannel signal y = {y gm , g = 1, . . . G, m = 1, . . . M }, g denoting the time index and M the channel index. Consider an orthonormal basis U = {u g , g = 1, . . . G} (here, g labels the atoms of the basis) for the single channel signal space. We are interested in expansions of the form y = g x g u g (where multichannel vectors are denoted with bold symbols), in cases where the observations are noisy, and the basis U has been chosen in such a way that the coefficient sequences x are sparse in the g direction, and persistent across channels. Then, we are close to the case described in Section 3.1.

In this case, A is an orthogonal matrix and the optimization problem is formulated as before:

min x∈R N ×M y -Ax 2 2 + λ x q p,q ,
M being the number of channels. Since the matrix A remains orthogonal, the results above may then be applied directly.

Since we aim at privileging groups of coefficients (persistence across channels), we choose the G-LASSO estimate provided in Proposition 1-(b). We illustrate this problem with a sound example recorded in a running train.

The considered signal features low frequency noise, phone ringings, voice, clicks and additional transient components. The signal is a four channels signal, recorded using three directional and one omni-directional microphones. Gaussian white noise was added to the four channels, yielding input SNR equal to 6 dB. The signal was denoised by applying LASSO (corresponding to the 1 norm prior on the set of coefficients), and G-LASSO (corresponding to 2,1 norm prior on coefficients). As stressed before, this choice is motivated by the desire of using the same significance map (i.e. the set of labels of nonzero coefficients) for all channels. Simulations were run with various values of the threshold (i.e. the Lagrange parameter). Corresponding SNR curves are displayed in Fig. 8.

The mixed norm based approach clearly outperforms the 1 norm approach significantly. Similar results (not shown here) were also obtained on different multichannel audio signals. The improvement appears to increase with the number of channels, as may be expected. In the particular example considered here, we remark that even though the four microphones are different (three being directional), the four signals are coherent enough for G-LASSO to improve significantly the LASSO results.

Let us finally stress that the same approach may be developed in many other multichannel signal denoising contexts, such as color image denoising, multispectral imaging,...

Denoising a "vinyl recording" like noisy signal

The E-LASSO and AE-LASSO estimates are now compared to the LASSO in the context of single channel denoising. In the standard additive Gaussian white noise benchmark, the soft-thresholding provided by LASSO is a better choice than the generalized shrinkage operators obtained using mixed-norms. However, based on Equation [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF], we also remark that E-LASSO and AE-LASSO are valuable alternatives when going beyond the Gaussian white noise assumption, in cases where the noise variance varies with the group index g (this was already visible in the experiments of Section 3.2).

Here we consider the case of (single channel) audio signal, perturbed by additive non-stationary noise, whose variance varies significantly with time. The considered example was taken from vinyl recordings1 . Vinyl recording noise (including many "cracks" and other nonstationary noises) was added to a musical signal (excerpt of about 6 s, 2 18 samples at 44100 Hz sampling rate, of the song "Mamavatu" from Susheela Raman), the resulting input SNR being about 1 dB only. This noisy signal was then expanded in a MDCT basis (with 512 samples -about 11 ms-long windows). The group index g (see Section 2) was chosen as the time index of the MDCT basis functions, and m as the frequency index. aim at extracting transient and tonal components. This problem has received increasing interest recently, as such a separation proves useful both by itself (for example for denoising [START_REF] Févotte | Sparse linear regression with structured priors and application to denoising of musical audio[END_REF] or compression [START_REF] Molla | An hybrid audio scheme using hidden Markov models of waveforms[END_REF], and as pre-processing step for various sound processing tasks (such as, signal analysis, for which different layers are analyzed using different approaches, or sound transformation, e.g. pitch shifting, for which the transformation has to be different for different layers). An intrinsic difficulty is the absence of ground truth that could be used for validating the proposed approaches. However, we shall see that the approach developed here is suitable for multilayered signal decomposition, and that different choices for the coefficient priors yield significantly different results.

Multilayered separation may be performed using various approaches [START_REF] Févotte | Sparse linear regression with structured priors and application to denoising of musical audio[END_REF][START_REF] Molla | An hybrid audio scheme using hidden Markov models of waveforms[END_REF] (see also the MCA algorithm for cartoon + texture separation in images [START_REF] Starck | Image decomposition via the combination of sparse representations and a variational approach[END_REF]). Here we illustrate the influence of the mixed-norm in the regression problem [START_REF] Fornasier | Recovery algorithm for vectorvalued data with joint sparsity constraints[END_REF], in comparison to the usual 1 norm used in the MCA regression problem [START_REF] Bobin | Morphological component analysis for sparse multichannel data: Application to inpainting[END_REF].

We choose a musical signal taken from the "Mamavatu" song (see above), that involves percussive instruments, voice and guitar. The signal duration is about 6 s (2 18 samples). Keeping the notations of subsection 2.3, one then expects to obtain an estimate V xV of the transient layer, and an estimate U xU of the tonal layer. We compare the estimates given by choosing two 1 norms (as in MCA), and several mixed norms, to be specified below. We choose for U a MDCT basis with a 4096 samples window length, and for V a MDCT basis with a 128 samples window length. The representations of the MDCT coefficients of the tonal (resp. transient) layer in U (res. V) are shown in Fig. 10. The particular struc- tures of both layers, with their persistence properties, appear clearly there.

The tonal layer is expected to be sparsely represented in the frequency domain, with emergent frequencies that may evolve slowly with time (i.e. almost horizontal lines of large MDCT coefficients). Possible choices for the estimates are E-LASSO (sparse within group) with the time label as group label, or G-LASSO (sparse across groups) with the frequency label as group label. However, the latter choice turns out to be a poor strategy for the tonal layer, because of the slow evolution in time of the frequencies. Furthermore, experiments show that for this example, LASSO and E-LASSO performs very similarly to estimate the tonal layer. We thus limit the present illustration to LASSO estimates for the tonal layer.

In a similar spirit, the transient layer is expected to be sparse in time, but spread out in the frequency domain. Then, for the transient layer, while E-LASSO with the frequency label as group label still seems a relevant choice, G-LASSO (with the time label as group label) is also interesting because of the particular structures of transients, which are most often sharply timelocalized.

Then, in order to show the differences between the 1 , 12 and 21 mixed-norms, we fixed the 1 norm to estimate the tonal layer, and we compared the results with the three different choices for the transient layer.

In the numerical simulations presented here, the λ and µ parameters were tuned to obtain approximately the same number of coefficients for each functional, to well illustrate the differences between the norms. Table 1 summarizes the results obtained using the three possible functionals. The first line of the table gives the choices that were made for norms for the tonal layer and the transient layer. The second and the third (resp fourth and fifth) lines give respectively the numbers of retained coefficients for x U (resp x V ) and SNR of this layer, defined as 20 times the base two logarithm of the ratio of the energy of the signal by the energy of the layer. The last two lines give the total number of retained coefficient and the SNR of the reconstruction x U + x V , defined as above. Here, SNRs should not be interpreted as a performance measure, but rather as a way to compare the behaviors of the three estimates. Hence, one can see that with the LASSO/LASSO choice, the transient layer is closer to the original signal than with the other two choices, but does not yield the best expected results for this layer (see figures and discussion below).

Together with this table, Fig. 11 and 12 clearly show the different behaviors of the estimators. Obviously, E-LASSO promotes persistence in comparison to LASSO. For the transient layer, the vertical structures are better preserved by E-LASSO. In comparison, the LASSO transient estimate catches a lot of low frequency components, which is generally not desirable.

The G-LASSO transient estimate (Fig. 12) performs quite differently. Like E-LASSO, it is not affected by low frequency components. In addition, it provides a very simple map of nonzero coefficients, which may be interesting for some tasks such as transient or onset detection. However, this estimate may also be considered an over simplification of the transient layer.

Even though the reconstructions obtained with the three decompositions are very similar (in terms of SNR and listening2 ), the behaviors of the layers are completely different. With the LASSO/LASSO choice, the low frequencies are shared between the two layers, while the partials are better preserved in the tonal layer with the LASSO/E-LASSO and LASSO/G-LASSO estimates. Fig. 11 shows the differences between the time-frequency coefficients for the three estimates of the tonal layer.

We also tried to replace E-LASSO estimate by its AE-LASSO approximation (17) (even though the convergence proof of Algorithm 1 is not valid any more in this case, we always observed numerical convergence). Let us stress that the main shortcoming of E-LASSO is the sensitivity to the regularization parameters λ and µ: slight changes to the parameters may affect the solution significantly. AE-LASSO appears to be much less sensitive to the choice of the regularization parameter.

Conclusion

We have shown in this paper the relevance of mixed norm priors in the framework of sparse regression problems. Such mixed norms have been extensively used in the mathematical analysis literature, but their use in practical situations are limited to some particular ones such as the G-LASSO and in the context of joint spar-sity for multichannel signals. For the sake of simplicity, the mixed norms discussed here are the 1,2 and 2,1 norms, but similar results may be obtained using more general p,q norms, and several standard sparse approximation algorithms may be extended to that situation. We refer to the forthcoming paper [START_REF] Kowalski | Sparse regression using mixed norms[END_REF] for a thorough analysis of the latter.

Mixed norms yield generalised shrinkage operators; we also proposed new generalizations of the latter, that allow one to refine signal modelling, and overcome some shortcomings of standard thresholding operations. The E-LASSO estimate (and its approximation AE-LASSO) is a solution for the "over-sparsifying" behavior of the 1 norm. The WG-LASSO is a valuable alternative to G-LASSO when no well-defined group of coefficients is available. We applied these operators on simulated signal to illustrate as clearly as possible their respective behaviors.

Here, we have only emphasized a couple of applications, in the domain of audio signal processing, for which the results were encouraging. Let us stress that our point was not to compare to state of the art approaches, but rather to show what can be done using very simple techniques, that can be refined further. We would also like to point out that this approach is not at all specific to audio signals, and may be applied mutatis mutandis to image decomposition, for example in the framework of the MCA approach of [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF], or multichannel signals such as EEG/MEG signals.

To conclude, it is worth coming back to the behavior of mixed norms in the present context. The rationale of our approach is to use a combination of 1 and 2 norms, to promote sparsity in the direction of one of the two indices, and persistence in the direction of the other. Now, as we have stressed at the beginning of this paper, a doubly labelled coefficient sequence can be obtained by arbitrary re-labelling of a given coefficient sequence. Therefore, mixed norm approaches can be used to introduce models for coefficients involving a small number of clusters of significant coefficients. Such a representation features both sparsity (in the domain of coefficient groups) and persistence (within a group). We believe that the potential of such approaches can be very important in a number of practical situations.

Fig. 1

 1 Fig. 1 Time-Frequency map with 8.5% non-zero coefficients, generated using fixed frequency Markov chains.

Fig. 2

 2 Fig. 2 An example of coefficient re-labelling.

Fig. 3

 3 Fig. 3 An example of sliding window.

Fig. 4

 4 Fig. 4 Comparison between LASSO, 2 types of G-LASSO and WG-GLASSO.

Fig. 5

 5 Fig. 5 Comparison between LASSO, 2 types of G-LASSO and WG-GLASSO.

Fig. 6

 6 Fig. 6 Comparison between LASSO, E-LASSO, AE-LASSO (approximation of E-LASSO), PE-LASSO (E-LASSO with persistence) and WG-LASSO; input SNR=3dB.

Fig. 7

 7 Fig. 7 Comparison between LASSO, E-LASSO, AE-LASSO, PE-LASSO and WG-LASSO; input SNR=5dB.

Fig. 8

 8 Fig. 8 Multichannel denoising of the train signal. SNR as a function of the number of retained coefficients; full curve: LASSO; dashed curve: G-LASSO; dashed-dotted curve: WG-LASSO.

Fig. 9

 9 displays the evolution of the output SNR as a function of the number of non-zero coefficients for the LASSO, E-LASSO, AE-LASSO, PE-LASSO and WG-LASSO. The behavior of the curves is close to the one observed on Fig.6 and 7of Section 3.2, even though the SNR improvement is not large.4.3 Multilayered audio signal expansionWe now consider the problem of decomposing (single channel) signals into layers of different nature, focusing again on the case of audio signals, from which we

Fig. 9

 9 Fig. 9 Comparison between LASSO, E-LASSO, AE-LASSO, PE-LASSO and WG-LASSO, on the single channel audio signal with additive "vinyl recording type" noise.

Fig. 10

 10 Fig. 10 MDCT coefficients of th signal. Top: in U, bottom: in V.

Fig. 11

 11 Fig. 11 MDCT coefficients of the three estimates of the tonal layer. From top to bottom: LASSO/LASSO, LASSO/E-LASSO, LASSO/G-LASSO

Fig. 12

 12 Fig. 12 MDCT coefficients of three estimates of the transient layer. From top to bottom: LASSO/LASSO, LASSO/E-LASSO, LASSO/G-LASSO.

Table 1

 1 Results obtained for three different choices of estimates. number of retained coefficients in each layer and reconstruction, and corresponding SNR values. L stands for LASSO, G-L for G-LASSO and E-L for E-LASSO.

	norms	L / L	L / EL	L / GL
	nbcoeff x U	16.4%	16.5%	16.7%
	SNR x U	7.6 dB	17.8 dB	20.2 dB
	nbcoeff x V	7.4%	7.5%	7.4%
	SNR x V	2.8 dB	0.22 dB	0.12 dB
	nbcoeff x U + x V	23.8%	24.1	24.2%
	SNR x V + x V	26.1 dB	25.4 dB	24.1 dB

Samples of vinyl recordings noise are available at the web site www.universal soundbank.com/audio.htm

Soundfiles of the different estimates can be listened from the website [1].

Acknowledgements

We wish to thank the anonymous referees for their comments, and for bringing a few additional references to our attention. We also thank Stéphane Molla for kindly providing the train sound example.


Badr Abdullah

Saeed Al-Tayar

Mon Mon Tin-Oo

Mohd Zulkarnain Sinor

Mohamed Hussein Abdikarim

Mohammed Sultan Al-Ak'hali

Oral melanosis and severe periodontitis in toombak Object 2 Search in any field users: a case report

Keywords: Toombak, smokeless tobacco, melanosis, periodontitis, case report

Background: Toombak is a traditional type of smokeless tobacco used by people in Yemen, especially in Dawan Valley, Hadramout Province, Yemen. The use of smokeless tobacco may be considered as a predisposing risk factor for several oral diseases.

Case Presentation: This paper presents two cases. The first is that of a 27-year-old male toombak user with severe periodontitis and oral melanosis. The second case is that of a 44year-old male toombak user with oral hypermelanosis. The patients were subjected to dental cleaning. Quit toombak counseling was conducted, and the patients were educated about the adverse effects of toombak usage.

Conclusion:

Oral tissues are severely affected by toombak usage. Thus, comprehensive toombak prevention programs should be implemented to reduce toombak-associated diseases.

dried powdered tobacco leaves mixed with ash [START_REF] Al-Tayar | Prevalence and association of smokeless tobacco use with the development of periodontal pocket among adult males in Dawan Valley, Yemen: a cross-sectional study[END_REF] ( Figure 1). Toombak is usually placed in the buccal lower or labial vestibules [START_REF] Salem | Leukoplakia and tobacco habits in Gizan, Saudi Arabia[END_REF], as shown in Figure 2. The association between traditional types of smokeless tobacco in Yemen and the development of oral diseases has been rarely considered.

Al-Tayar et al. presented the periodontal status of 346 Yemenis in Dawan Valley, Yemen. In their study, multiple logistic regression analysis revealed that regular toombak users were 6.62 times more likely to develop periodontal pocket than those who had never used it [START_REF] Al-Tayar | Prevalence and association of smokeless tobacco use with the development of periodontal pocket among adult males in Dawan Valley, Yemen: a cross-sectional study[END_REF].

As regards toombak-induced alterations in oral mucosal tissues, Al-Tayar et al. [START_REF] Ba | Association between Shammah use and oral leukoplakia-like lesions among adult males in Dawan Valley, Yemen[END_REF] found that the participants who were using toombak at the time were almost 13 times more likely to develop oral leukoplakia-like lesions than those who had never used it. In terms of oral mucosal melanosis, several studies have confirmed the association between smoking and such lesion [START_REF] Tadakamadla | Effect of smoking on oral pigmentation and its relationship with periodontal status[END_REF] but not with smokeless tobacco. However, Naveen-Kumar et al. [START_REF] Naveen-Kumar | Various forms of tobacco usage and its associated oral mucosal lesions[END_REF] observed tobacco-associated melanosis most frequently in conventional smokers, followed by reverse smokers and smokeless tobacco group. Current clinical reports describe various intraoral findings that suggest the oral mucosal changes and the degenerative and destructive magnitude of periodontal tissues in patients with toombak use history of more than 8 years.

Case Presentation

The first case is of a 27-year-old male who visited Al-Ebtessamah Dental Clinic in Dawan

Valley, Yemen, with the chief complaint of pain in the lower gum from the right to left canine region. The patient was a toombak user for more than 8 years. His habitual placement of toombak was on the lower right muco-buccal vestibule and sometimes on the lower labial vestibule. Clinical examinations showed severe deposition of dental calculus and grade III mobility of mandibular left central incisor and mandibular right canine. Periodontal probing revealed a pocket depth ranging from 6 to 10 mm in the mandibular anterior region. Oral mucosal pigmentation was found on the site of habitual toombak placement ( Figure 3). The case was diagnosed as chronic periodontitis and oral mucosal melanosis induced by extraction was planned for the mandibular left central incisor and mandibular right canine. The patient was also subjected to planned sessions of scaling and root planning and was instructed to maintain good oral hygiene. Analgesic and chlorhexidine mouthwash were prescribed.

Quit toombak counseling was conducted, and the patient was educated about the adverse effects of toombak usage.

The second case is of a 44-year-old male who presented to the Al-Ebtessamah Dental Clinic in Dawan Valley, Yemen, with a complaint of mandibular anterior teeth discoloration with black pigmentation on the lower anterior muco-labial vestibule ( Figure 4).

Discussion

The reported cases involved advanced loss of periodontal tissues obviously related to toombak usage of more than 8 years. A possible contributing mechanism in the development of severe periodontal disease is the cholinergic effect of toombak along with the calcium salt in the saliva produced during hypersalivation causing calculus deposition. The heavy calculus deposition can then destroy periodontal tissue in smokeless tobacco users [START_REF] Ling | Association between betel quid chewing, periodontal status and periodontal pathogens[END_REF]. This finding is confirmed by a case report presented by Giri et al. [START_REF] Giri | Betel nut chewing associated with severe periodontitis[END_REF], which showed that smokeless tobacco in the form of betel nut chewing is associated with advanced periodontal tissue loss. The present case report is also supported by the study of Parmar et al. [START_REF] Parmar | Effect of chewing a mixture of areca nut and tobacco on periodontal tissues and oral hygiene status[END_REF], which revealed the significantly higher development of periodontal pockets and gingival recession in smokeless tobacco users than in non-users.

The color change of the oral mucosa is due to the accumulation of one or more tissue pigments. The pigments associated with mucosal discoloration could be classified as endogenous, such as melanin pigments, and exogenous, such as drug-related pigments.

Melanin-associated lesions are the most common pigmentations. Both the smoked and smokeless forms of tobacco induce oral mucosal changes, among which intraoral mucosal pigmentation is a clinical manifestation [START_REF] Mirbod | Tobacco-associated lesions of the oral cavity: Part I. Nonmalignant lesions[END_REF]. Tobacco use, especially smoking, may augment melanin production, which may provide a biological defense against the noxious agents present in tobacco smoke [START_REF] Hedin | Melanin depigmentation of the palatal mucosa in reverse smokers: a preliminary study[END_REF]. Our report stated that toombak may induce oral melanosis.

This result is supported by the report of Kumar and Chaturvedi [START_REF] Kumar | Oral melanosis[END_REF], which revealed brown pigmentation in the ventral aspect of the tongue of a patient who used smokeless tobacco, specifically, chewing tobacco, for 6 years.

Conclusion

Toombak affects the oral tissues of its users. Thus, comprehensive toombak prevention programs should be implemented to reduce toombak-associated diseases.

medications for the second case.

Clinical

Figure 1 :

 1 Figure 1: Toombak.Figure

Figure 2 :

 2 Figure 1: Toombak.Figure 2: Toombak in the lower labial vestibule of the oral cavity.

  Toombak use history was 8.5 years. The pigmentation was found on the lower anterior muco-labial vestibule where the patient typically placed the toombak. The changes were asymptomatic. This pigmentation was clinically diagnosed as oral melanosis aggravated by long-term toombak usage. The patient was subjected to scaling and polishing. Quit toombak counseling was conducted, and the patient was educated about the adverse effects of toombak usage.

Figure 3 :

 3 Figure 3: Chronic periodontal destruction and changes of

Figure 4 :

 4 Changes of oral melanosis at lower anterior muco-labial vestibule.

  Mohammed Sultan Al-akhali. "Oral melanosis and severe periodontitis in toombak users: a case report." European Journal of Medical Case Reports 2.January 2018 (2018), 9-11.Accordingly, the Publisher and the Editorial Boards disclaim all responsibilities for the consequences of any such inaccurate or misleading data, opinion, or statement. To the extent permissible under applicable laws, no responsibility is assumed by the Publisher nor by the Editors for any injury and/or damage to persons or property as a result of any actual or alleged libellous statements, infringement of intellectual property or privacy rights, or products liability, whether resulting from negligence or otherwise, or from any use or operation of any ideas, instructions, procedures, products or methods contained in the material therein. Whilst every effort is made to ensure that drug doses and other quantities are presented accurately, readers are advised that new methods and techniques involving drug usage, and described in DPG journals, should only be followed in conjunction with the drug manufacturer's own published literature in their own country.

	Badr Abdullah Al-Tayar, Mon Mon Tin-Oo, Mohd Zulkarnian Sinor, Mohamed Hussein • Privacy & Cookies Close
	Procedure Abdikarim, Mohammed Sultan Al-akhali. Oral melanosis and severe periodontitis in toombak 5 • Sitemap (XML) Extraction was planned for the mandibular left central incisor and users: a case report. EJMCR. (2018), [cited January 14, 2020]; 2(January 2018): 9-11. Print. doi:10.24911/ejmcr/1/35 mandibular right canine, and scaling and root planning for the first case Download doi:10.24911/ejmcr/1/35 and scaling and polishing for the second case • Sample CARE-checklist
	Specialty • CARE-checklist 6 Dental Public Health
	APA (American Psychological Association) Style • Patient Consent Form (WORD doc)
	How to Cite this Article Harvard Style Badr Abdullah Al-Tayar, Mon Mon Tin-Oo, Mohd Zulkarnian Sinor, Mohamed Hussein • Patient Consent Form (pdf)
	Pubmed Style Badr Abdullah Al-Tayar, Mon Mon Tin-Oo, Mohd Zulkarnian Sinor, Mohamed Hussein Badr Abdullah Al-Tayar, Mon Mon Tin-Oo, Mohd Zulkarnian Sinor, Mohamed Hussein Abdikarim, Mohammed Sultan Al-akhali (2018) Oral melanosis and severe periodontitis in Policies Abdikarim, Mohammed Sultan Al-akhali (2018) Oral melanosis and severe periodontitis in toombak users: a case report. EJMCR, 2 (January 2018), 9-11. doi:10.24911/ejmcr/1/35 toombak users: a case report. European Journal of Medical Case Reports, 2 (January 2018), 9-11. • Editorial Policies doi:10.24911/ejmcr/1/35 • Publication Ethics and Publication Malpractice Statement
	Abdikarim, Mohammed Sultan Al-akhali. Oral melanosis and severe periodontitis in toombak • Conflict of Interest Policy
	• Plagiarism Policy
	• Protection of Research Participants Turabian Style Journal • Website Terms & Conditions
	Badr Abdullah Al-Tayar, Mon Mon Tin-Oo, Mohd Zulkarnian Sinor, Mohamed Hussein • Thank you to our reviewers
	Abdikarim, Mohammed Sultan Al-akhali. 2018. Oral melanosis and severe periodontitis in • Editorial & Peer Review Process
	toombak users: a case report. European Journal of Medical Case Reports, 2 (January 2018), 9-11. • Publishing Ethics
	doi:10.24911/ejmcr/1/35 • Abstracting & indexing
	• Reviewer Login
	doi:10.24911/ejmcr/1/35 • About Publisher
	Chicago Style Authors
	AMA (American Medical Association) Style Badr Abdullah Al-Tayar, Mon Mon Tin-Oo, Mohd Zulkarnian Sinor, Mohamed Hussein Abdikarim, Mohammed Sultan Al-akhali. Oral melanosis and severe periodontitis in toombak • Instructions for authors Badr Abdullah Al-Tayar, Mon Mon Tin-Oo, Mohd Zulkarnian Sinor, Mohamed Hussein • Author's Rights and Obligations Abdikarim, Mohammed Sultan Al-akhali. "Oral melanosis and severe periodontitis in toombak • Self-Archiving Policies users: a case report." European Journal of Medical Case Reports 2 (2018), 9-11. doi:10.24911/ejmcr/1/35 • Submit an article
	users: a case report. EJMCR. 2018; 2(January 2018): 9-11. doi:10.24911/ejmcr/1/35 • Track your Article
	Help
	• Contact us
	Vancouver/ICMJE Style • Disclaimer
	• Advertising

users: a case report. EJMCR. 2018; 2(January 2018): 9-11. doi:10.24911/ejmcr/1/35 Web Style Badr Abdullah Al-Tayar, Mon Mon Tin-Oo, Mohd Zulkarnian Sinor, Mohamed Hussein Abdikarim, Mohammed Sultan Al-akhali. Oral melanosis and severe periodontitis in toombak users: a case report. https://www.ejmcr.com/?mno=277685 [Access: January 14, 2020].

MLA (The Modern Language Association) Style

Badr Abdullah Al-Tayar, Mon Mon Tin-Oo, Mohd Zulkarnian Sinor, Mohamed Hussein Abdikarim,

Disclaimer

×

Although every effort is made by the Editorial Board of the concerned journals and the publisher (Discover Publishing Group) to see that no inaccurate or misleading data, opinion or statement appear in the journal(s) or websites managed or published by Discover Published Group, we want to make it clear that the data and opinions appearing in the articles and advertisements herein are the responsibility of the contributors, sponsors, or advertisers concerned.

Acknowledgement

The authors acknowledge the staff at the Al-Ebtesammah Dental Clinic in Dawan Valley, Yemen, who helped to collect patient data and images.

List of Abbreviations

None

Consent for publication

Informed consent was obtained from the patients to publish this case.

Ethical approval

Ethical approval is not required at our institution for publishing a case report in a medical journal.

Author details

Badr Abdullah Saeed Al-Tayar


Leonardo Tomazeli Duarte
email: leonardo.duarte@gipsa-lab.inpg.fr

Christian Jutten
email: christian.jutten@gipsa-lab.inpg.fr

A NONLINEAR SOURCE SEPARATION APPROACH FOR THE NICOLSKY-EISENMAN MODEL

, we proposed source separation methods for a simplified version of the Nicolsky-Eisenman (NE) model, which is related to a chemical sensing application. In the present paper, we provide a method able to deal with the complete NE model. Basically, such a model can be seen as a composition of a non-diagonal nonlinear transformation followed by a diagonal nonlinear transformation, i.e. a set of component-wise functions. The basic idea behind the developed technique is to estimate the parameters of these two stages in a separate fashion by using a prior knowledge of the sources, namely the fact that one of the sources is constant during a certain period of time. Simulations attest the viability of the proposed technique.

INTRODUCTION

The problem of blind source separation (BSS) concerns the retrieval of an unknown set of source signals by using only samples that are mixtures of these original signals. There is a great number of methods for the situation in which the mixing process is linear. However, it seems that in some applications the linear approximation is not suitable and, as a consequence, it becomes necessary to design source separation methods that take the nonlinearity of the model into account. The main problem in the nonlinear case is that the ubiquitous tool for performing source separation, the independent component analysis (ICA), does not work in a general situation [START_REF] Hyvärinen | Nonlinear independent component analysis: existence and uniqueness results[END_REF][START_REF] Jutten | Advances in blind source separation (BSS) and independent component analysis (ICA) for nonlinear mixtures[END_REF]. In other words, there are cases in which the recovery of the statistical independence, which is the very essence of ICA, does not guarantee the separation of the sources when the mixture model is nonlinear. In view of this limitation, a more reasonable approach is to consider constrained mixing systems as, for example, post-nonlinear (PNL) mixtures [START_REF] Jutten | Advances in blind source separation (BSS) and independent component analysis (ICA) for nonlinear mixtures[END_REF] and linear-quadratic mixtures [START_REF] Hosseini | Blind separation of linearquadratic mixtures of real sources using a recurrent structure[END_REF].

In recent papers [START_REF] Bedoya | Improving semiconductor-based chemical sensor arrays using advanced algorithms for blind source separation[END_REF][START_REF] Duarte | A mutual information minimization approach for a class of nonlinear recurrent separating systems[END_REF], the problem of BSS in a particular class of nonlinear systems related to a chemical sensing application was investigated. For instance, we developed in [START_REF] Duarte | Blind source separation of a class of nonlinear mixtures[END_REF][START_REF] Duarte | A mutual information minimization approach for a class of nonlinear recurrent separating systems[END_REF] source separation methods for estimating the concentrations of ions in a solution when the valences of these ions are different. In our first works, we considered a simplified version of the mixing model that is pertinent only in the situations where the sensors present a Nernstian response [START_REF] Umezawa | Potentiometric selectivity coefficients of ion-selective electrodes[END_REF]. In the present work, by relying on an additional assumption on the sources, we extend our solution to the complete model, which permits us to work in more realistic situations. Our work will be presented as follows: in Section 2 we introduce the problem. After that, in Section 3, we expose the basics of our method. In Section 4, simulations are carried out in order to illustrate the viability of the proposal. Finally, in Section 5, we state our conclusions and perspectives.

PROBLEM STATEMENT

In analytical chemistry, a relevant problem is to estimate the evolution of the concentrations of several ions in a solution. An inexpensive and practical approach is based on the use of potentiometric sensors, such as glass-electrodes and ionsensitive field-effect transistors (ISFET). Basically, the sensing mechanism in this sort of sensor is due to the dependence of the potential difference between two electrodes on the concentration of a target ion [START_REF]Microcapteurs chimiques et biologiques[END_REF]. Unfortunately, there is a pronounced lack of selectivity behind this principle, i.e. the generated potential difference in the sensor can also depend on other interfering ions present in the solution.

A very common approach to deal with the interference problem is to perform a calibration of the chemical sensor by using samples from a known database [START_REF]Microcapteurs chimiques et biologiques[END_REF]. However, there are two major difficulties in such an approach: 1) the acquisition of training samples may be time-demanding and 2) this calibration procedure must be performed from time to time due to the sensors drift. In this context, an approach founded on array of chemical sensors and on BSS methods emerges as an attractive alternative, since, in this case, the estimation of the evolution of the concentrations can be done by using only the sensors responses, that is, without conducting supervised learning with a known data set.

Although very general, BSS techniques usually require a parametric description of the mixing model. In our case, this information can be found on the classical model of Nicolsky-Eisenman (NE) [START_REF]Microcapteurs chimiques et biologiques[END_REF], which provides a simple and yet adequate description of the potentiometric sensors. According to this model, the response of the i-th sensor is given by:

x i (t) = c i + d i log s i (t) + ∑ j, j =i a i j s j (t) z i z j , (1)
where t corresponds to the time index; s i (t) and s j (t) denote the concentration of the target ion and of the concentration of the j-th interfering ion, respectively. z i and z j denote the valence of the ions i and j, respectively. The selective coefficients a i j model the interference process; c i and d i are constants that depend on some physical parameters. Note that when the ions have the same valence, then the model (1) can be seen as a particular case of the class of PNL systems, as described in [START_REF] Bedoya | Improving semiconductor-based chemical sensor arrays using advanced algorithms for blind source separation[END_REF].

In this work, we are interested in the situation where the valences are different (z i = z j ). In view of the toughness of the resulting model, we investigate only the case with two sources. Also, the parameters c i may be not considered, since they only introduce an ambiguity with respect to the sources mean value. Therefore, the obtained mixing model can be described as:

x 1 (t) = d 1 log p 1 (t) = d 1 log s 1 (t) + a 12 s 2 (t) k x 2 (t) = d 2 log p 2 (t) = d 2 log s 2 (t) + a 21 s 1 (t) 1 k , (2)
where k = z 1 /z 2 . As illustrated in Fig. 1, this mixing model is composed of two stages, being the first one a particular nonlinear mapping that depends on the valences of the target ions and the second one a pair of nonlinear component-wise functions parametrized in d i . As it will be clarified in the sequel, the proposed method treats these two stages in a separate way.

PROPOSED METHOD

A natural strategy to perform BSS on the model ( 2) is to adopt the two-stage separating system shown in Fig. 1. The first stage is composed of exponential component-wise functions exp(x i /d * i ), whereas the second one should provide a non-diagonal mapping able to counterbalance the effects introduced by the first stage of the mixing system. In [START_REF] Duarte | Blind source separation of a class of nonlinear mixtures[END_REF][START_REF] Duarte | A mutual information minimization approach for a class of nonlinear recurrent separating systems[END_REF], we investigated source separation methods for this first stage and we assumed that the component-wise functions were known in advance. In fact, this situation is realistic only when the sensors have a Nernstian response, given that in such a case the parameters d i are known in advance.

In the sequel, we will present a brief review of our previous works. Then, we will detail the main contribution of this paper which is related to the estimation of the componentwise functions, i.e., of the parameters d i . With this new step, it becomes possible to develop a complete source separation method for the model described in [START_REF] Babaie-Zadeh | A geometric approach for separating post non-linear mixtures[END_REF]. During our development, the following assumptions are considered: In the context of the ion sensing application, assumption 1 is equivalent to consider that there is no chemical reaction between the ions. Concerning the assumptions 2 and 3, our separation methods would work in a predefined range of concentrations, as is usual in commercial sensors. Furthermore, it is quite natural to consider positive sources, since they represent concentrations. Finally, assumption 4 state a knowledge of the ions' valences. Actually, this information is available since the main interfering ions, and so their valences, known in advance.

Review of our solution for a simplified version of (2)

When the component-wise functions in (2) are known in advance, the mixing model becomes

p 1 (t) = s 1 (t) + a 12 s 2 (t) k p 2 (t) = s 2 (t) + a 21 s 1 (t) 1 k . (3)
In order to retrieve s i , we adopted [START_REF] Duarte | Blind source separation of a class of nonlinear mixtures[END_REF] the following recurrent network as separating system:

y (n + 1) = p 1 (t) -w 12 y 2 (n) k y 2 (n + 1) = p 2 (t) -w 21 y 1 (n) 1 k , (4)
where the vectors w = [w 12 w 21 ] T and y(n

) = [y 1 (n) y 2 (n)] T
denote the parameters to be adjusted and the system outputs at time n, respectively. For a given sample of the mixtures [p 1 (t) p 2 (t)] T , and for a given value w, the system outputs are obtained after the convergence of the dynamics (4).

In order to adjust the parameters w in (4), we proposed [START_REF] Duarte | A mutual information minimization approach for a class of nonlinear recurrent separating systems[END_REF] the following learning rule that minimizes the mutual information of the vector y:

w ← w -µE ∂ y ∂ w T β y (y) , (5)
where µ denotes the learning rate, β y (y) is the score function difference vector associated with the random variable y and ∂ y ∂ w is the Jacobian of (4) with respect to w.

An approach for inverting the component-wise functions

Now, let us turn our attention to the complete version of the NE model. The following parametric model can be considered in the first stage of the separating system

e 1 (t) = exp x 1 (t) d * 1 e 2 (t) = exp x 2 (t) d * 2 , (6)
where d * i are the parameters to be adjusted. A first natural approach to adapt the separating system would be to develop an ICA algorithm to find the parameters {d * 1 , d * 2 , w 12 , w 21 }. In other words, we could seek a set of parameters that optimizes a functional associated with the statistical independence, such as the mutual information. However, it would be tough to develop a gradient rule in this case given that the separating system is composed of a nonlinear dynamic system and, therefore, there would be a risk of instability during the learning phase. Besides this practical problem, there is a crucial theoretical point that should be addressed: is the considered mixing model separable, i.e., is the recovery of the statistical independence enough to assure source separation? We do not have such an answer, but recent studies [START_REF] Jutten | Advances in blind source separation (BSS) and independent component analysis (ICA) for nonlinear mixtures[END_REF] suggest that for nonlinear transformations like the one dealt with here (2), it is necessary to consider prior information other than independence to achieve source separation.

In view of the aforementioned problems, we believe that it is more reasonable to consider a two-step approach, in which the component-wise functions are firstly estimated and, then, the recurrent system is trained. The price to be paid is that we need the following additional assumption: Assumption 5 There is, at least, a period of time where one, and only one, of the sources has zero-variance.

In view of assumption 5, if we consider, for instance, that s 1 (t) takes a constant value S 1 (where S 1 ∈ [S min i , S max i ] ) in a given time window, then, according to (2), we have

p 1 (t) = S 1 + a 12 s 2 (t) k p 2 (t) = s 2 (t) + a 21 S 1 k 1 . (7)
From this expression, it is not difficult to see that

p 1 (t) = S 1 + a 12 (p 2 (t) -a 21 S 1 k 1 ) k . (8)
Hence, under such an assumption, ( 8) is a polynomial of order k in the (p 1 , p 2 ) plane. Our idea is based on this fact, in the sense that this polynomial function is lost after the application of the component-wise functions and, thus, we may invert the log functions by searching a set {d * 1 , d * 2 } that restores a polynomial in the (e 1 , e 2 ) plane. In the sequel, we shall detail this idea.

Firstly, let us describe the mapping between the (p 1 , p 2 ) and (e 1 , e 2 ) planes. By considering ( 2) and ( 7), one can rewrite [START_REF] Deville | Temporal and time-frequency correlation-based blind source separation methods. part i: Determined and underdetermined linear instantaneous mixtures[END_REF] as:

e 1 (t) = exp d 1 log(p 1 (t)) d * 1 = p 1 (t) d 1 d * 1 e 2 (t) = exp d 2 log(p 2 (t)) d * 2 = p 2 (t) d 2 d * 2 . ( 9
)
The next step of our study is to find the relation between the data in the (e 1 , e 2 ) plane. As stated by [START_REF] Duarte | A mutual information minimization approach for a class of nonlinear recurrent separating systems[END_REF], there is a polynomial relation in the (p 1 , 2 ) plane, which can be expressed by

p 1 (t) = k ∑ i=0 ϕ i p 2 (t) i , (10)
where the coefficients ϕ i can be determined by the binomial expansion of (8). After a straightforward development considering ( 9) and ( 10), the following expression is obtained:

e 1 (t) = k ∑ i=0 ϕ i e 2 (t) d * 2 d 2 i d 1 d * 1 . ( 11
)
As stated above, our initial idea to find d * 1 and d * 2 is based on the recovery of a polynomial curve of order k in the (e 1 , e 2 ) plane. In order to verify the viability of our approach, we need to investigate the following question: for what values of d * 1 and d * 2 the function ( 11) is a polynomial of order k? At a first sight, it is clear that when the optimum solution

(d * 1 = d 1 , d * 2 = d 2 )
is achieved, then that expression results in a polynomial curve in the (e 1 , e 2 ) plane.

However, there is a particular situation where a polynomial curve is obtained although the mapping ( 9) is still nonlinear. In fact, when S 1 is null, then we can see from ( 8) that all the coefficients except ϕ k in [START_REF] Hyvärinen | Nonlinear independent component analysis: existence and uniqueness results[END_REF] are null. In this situ- ation, the following solution

(d * 1 = Dd 1 , d * 2 = Dd 2 )
, where D is a constant, also gives a polynomial of order k although does not correspond to our desired solution. Indeed, one of the reasons behind assumption 2 is exactly to avoid this situation by considering only positive sources.

To implement our idea, we must define a way to check if a set of points in the (e 1 , e 2 ) plane corresponds to a polynomial of order k. This can be done by defining a cost function as the mean square of the residuals resulting from the regression of the set of sample {(e 1 (t), e 2 (t))} N t=1 (being N the number of samples) through a polynomial of order k. In mathematical terms, this cost function can be expressed as min

d * 1 ,d * 2 ∑ t e 1 (t) - k ∑ i=0 α i (e 2 (t)) i 2 , ( 12
)
where α i correspond to the i-th regression coefficient.

In order to gain more insight, let us substitute ( 11) in ( 12), which gives min

d * 1 ,d * 2 ∑ t    k ∑ i=0 ϕ i (e 2 (t)) d * 2 d 2 i d 1 d * 1 - k ∑ i=0 α i (e 2 (t)) i    2 . ( 13
)
One may note that this expression is a nonlinear function with respect to the parameters {d * 1 , d * 2 }. Moreover, according to equation [START_REF] Hyvärinen | Nonlinear independent component analysis: existence and uniqueness results[END_REF], for a given sample at time t there is an underlying relation between e 1 (t) and e 2 (t), which in turn makes the regression coefficients α i nonlinearly dependent on the parameters to be optimized. As a consequence, it becomes difficult to obtain the derivatives of this function and, therefore, to develop a gradient-based optimization method.

An alternative approach to optimize (12) can be found on the so-called evolutionary techniques. Briefly, this class of algorithms performs a searching procedure that is based on the notion of population, i.e., it deals with a set of possible solutions (individuals). At each iteration, some perturbation (mutation and recombination, for instance) is introduced on the population and a group of individuals is selected to continue in the population of the next iteration (usually, the selected individuals are the ones with higher fitness, which is calculated through the cost function). The major benefit brought by an evolutionary method to our problem is that no information about the derivatives is needed, since the selection stage is based only on the evaluation of the cost function, which, for our problem, is a straightforward task.

Detection of zero-variance periods

The idea described in the last section works under the assumption that there is a time window in which one of the sources does not vary. Evidently, if a blind scenario is envisaged, then one should be able to detect this "silent period". A possible way to perform this task is to consider the problem from a geometric standpoint. Given that the mixing model is invertible and the sources are supposed bounded, the borders of the distribution in the (x 1 , x 2 ) plane corresponds to the situation in which one of the source is constant. Therefore, at least in an ideal situation, we could detect the silent periods by estimating the borders of the distribution of the mixtures, in the same way as performed in [START_REF] Babaie-Zadeh | A geometric approach for separating post non-linear mixtures[END_REF]. Note that this strategy works even when the assumption 5 is not met. Unfortunately, this procedure is difficult to implement since it demands a very accurate estimate of the borders, which may be difficult to achieve when the number of samples is small.

A second approach to search the silent periods is based on the fact that when one of the sources is constant, each sensor response corresponds to a deterministic function of the same random variable, i.e., x 1 = g 1 (s 2 ) and x 2 = g 2 (s 2 ). Therefore, we can argue that such situation is the one with maximum (nonlinear) correlation between the sensors and, given that, we may try to identify the silent periods by searching time windows for which a measure of correlation is maximized. This idea has already been developed for linear source separation [START_REF] Deville | Temporal and time-frequency correlation-based blind source separation methods. part i: Determined and underdetermined linear instantaneous mixtures[END_REF] and, in this case, the silent periods are found by observing the second-order correlation measure. In our case, we deal with a nonlinear model and, as consequence, a measure able to detect nonlinear dependences must be employed. With this purpose in mind, we consider the mutual information between the mixing signals.

The mutual information of two continuous random variables lies on the interval 0 ≤ I(x 1 , x 2 ) < +∞, being zero when x 1 and x 2 are statistically independent, and tending to infinity when there is a deterministic relation between these variables. Therefore, we can find the silent periods by looking at the time windows for which the mutual information is maximized. In fact, it seems more practical to maximize a normalized version of the mutual information defined as ς (x 1 , x 2 ) = 1 -exp(-2I(x 1 , x 2 )), given that its maximum value is one and occurs when there is a deterministic relationship between x 1 and x 2 .

To illustrate the idea of the last paragraph, we present in Fig. 2 the evolution of the normalized mutual information between the sensors response (estimated through a time window of length 151) and the respectively sources. Note that the maximum of the mutual information occurs exactly for time windows containing a constant source.

Description of the complete algorithm

We can summarize the complete separation algorithm in Tab.1. Concerning the first step, we adopted the mutual information estimator proposed in [START_REF] Darbellay | Estimation of the information by an adaptive partitioning of the observation space[END_REF]. As already discussed, the optimization of [START_REF] Jutten | Advances in blind source separation (BSS) and independent component analysis (ICA) for nonlinear mixtures[END_REF] in the second stage can be carried out through evolutionary methods. In this work, we chose the opt-aiNet algorithm. This evolutionary method has been proven to be efficient in signal processing applications (see [START_REF] Attux | Blind search for optimal wiener equalizers using an artificial immune network model[END_REF], for instance). In addition to its robustness to local minima, the opt-aiNet only needs zero-order information, which, as discussed before, is a very interesting feature for our problem. The technical details of this method can be found in [START_REF] De Castro | Learning and optimization using the clonal selection principle[END_REF].

SIMULATION RESULTS

To assess the performance of the algorithm described in Tab. 1, we simulated the problem of detecting the ions Ca 2+ and Na + through an array of two sensors (each one has a different ion as target). For that, we consider the parameters a 12 = 0.79 and a 21 = 0.40, which were taken from [START_REF] Umezawa | Potentiometric selectivity coefficients of ion-selective electrodes[END_REF]. Also, we have assumed that both sensors have a perfect Nerstian response [START_REF]Microcapteurs chimiques et biologiques[END_REF], i.e., d 1 = 0.0129 and d 2 = 0.0258. The efficacy of the obtained solution for each source was quantified according to the following index:

SIR i = 10 log E{s 2 i } E{(s i -y i ) 2 } . (14)
Thus, SIR = 0.5(SIR 1 + SIR 2 ) defines a global index.

Regarding the parameters of the algorithm, a set of 1000 samples was considered. The detection of the silent periods is performed by estimating the mutual information for a window of a length of 151 samples. Actually, it is difficult to achieve a reliable estimation of the mutual information with a smaller number of samples. Finally, concerning the training of the recurrent network, the number of iterations was 1500 and the learning rate was µ = 0.02.

Under the described scenario and with the two sources shown in Fig. 3, our method has achieved the following performance SIR 1 = 40.52, SIR 2 = 38.27 and SIR = 39.39. Such a result attests that our proposal does well in this case, as can be confirmed by looking at the mixing signals in Fig. 4 and at the retrieved sources in Fig. 5.

CONCLUSIONS AND PERSPECTIVES

In this work, we have proposed a source separation method for the NE model, which is related to a chemical sensing application. Under the assumption that one of the sources does not vary during a period of time, it became possible to estimate the component-wise functions, which correspond to the second stage of the NE model. By joining this stage to our previous solutions, we could define a complete separation framework. In order to verify the efficacy of our proposal, we conducted some experiments considering a set of parameters taken from the literature. The obtained solutions highlight that our method is a promising one to the application in mind.

There are several perspectives for this work. A first one is to investigate the extension of our method for the situation with more than two sources. Also, we intent to conduct some real experiments to investigate the applicability of our method in a real problem. By proceeding this way, we will be able to address several questions in more details as, for instance, the study of the noisy case, starting with the problem of modeling the noise in this sort of application.

Figure 1 :

 1 Figure 1: The Nicolsky-Eisenman model.

Figure 2 :

 2 Figure 2: Mutual information between the mixtures (a) and corresponding sources (b) (c).

Figure 3 :

 3 Figure 3: Sources.

  Figure 4: Mixtures.

Figure 5 :

 5 Figure 5: Retrieved sources.

Assumption 1

 1 The sources are statistically independent;

	Assumption 2 The sources are positive and bounded, i.e, s i (t) ∈ [S min i , S max i ], where S max i > S min i > 0;
	Assumption 3 The system expressed in (2) is invertible in the region given by [S min 1 , S max 1 ] × [S min 2 , S max 2 ];
	Assumption 4 k is known and takes only positive integer
	values;

Table 1 :

 1 Algorithm 1. Detection of silent periods• Estimate the mutual information between the mixtures x 1 and x 2 for a moving-time window. • Select the time window in which the mutual information is maximum. 2. Estimation of the component-wise functions.

	• For the selected time window, minimize ex-pression (12) with respect to d * 1 and d * 2 .
	3. Training the recurrent network
	• Determine the parameters w i j of (4) through
		the algorithm (5). The inputs of the recurrent
		network are e i .
		0.08
	[Ca 2+ ] (M)	0.04 0.06
		0.02
		0	0	200	400	600	800	1000
		0.1
	[Na + ] (M)	0.04 0.06 0.08
		0.02
		0	0	200	400	600	800	1000

Scientific and Technological Development (CNPq-Brazil) for funding his PhD research.