Elizabeth Pearce

Michael D Crossland

Gary S Rubin

The Efficacy of Low Vision Device Training in a Hospital based Low Vision Clinic

Keywords: Low vision, visual rehabilitation, quality of life

Background/Aims

Hospital based low vision services in the UK typically involve one consultation with an optometrist. In this study we investigate the effect of adding further low vision device training.

Methods

Participants were recruited from those attending their first low vision assessment (LVA). Participants completed the Massof Activity Inventory (MAI) questionnaire by telephone before their appointment. After LVA, participants were randomised to an intervention group (who received a further appointment to review device handling) or a control group. The MAI was readministered one and three months after the initial LVA. MAI data were converted to Rasch scores for goal difficulty.

Results 96 participants completed the study. Both groups experienced a significant improvement in goal difficulty following low vision intervention, (p<0.0001), equivalent to a visual acuity improvement of approximately 0.55 logMAR. There was no significant difference in improvement between the group randomised to the training visit and those in the control arm (p=0.80).

Conclusion

Self-perceived difficulty with daily visual tasks decreases following a low vision appointment. An additional visit for device handling training resulted in no further improvement. This could be due to the relatively simple nature of the devices prescribed in this clinic.

Introduction

Despite advances in treatment for sight threatening conditions, at least one in twenty-five people in the UK over the age of 40 have significant vision loss. [START_REF] Congdon | Causes and prevalence of visual impairment among adults in the United States[END_REF] Vision loss can have a profound impact on quality of life, 2 although this varies among individuals. [START_REF] Hazel | Visual function and subjective quality of life compared in subjects with acquired macular disease[END_REF] Visual impairment has the potential to impact upon a person's physical, emotional and social well-being, as well as their independence. [START_REF] Haymes | Relationship between vision impairment and ability to perform activities of daily living[END_REF][START_REF] Ramrattan | Prevalence and causes of visual field loss in the elderly and associations with impairment in daily functioning: the Rotterdam Study[END_REF][START_REF] Szlyk | Difficulty in performing everyday activities in patients with juvenile macular dystrophies: comparison with patients with retinitis pigmentosa[END_REF][START_REF] West | How does visual impairment affect performance on tasks of everyday life? The SEE Project. Salisbury Eye Evaluation[END_REF][START_REF] Williams | The psychosocial impact of macular degeneration[END_REF] The loss of sight can present practical barriers to common daily activities such as reading. [START_REF] Mcclure | Macular degeneration: do conventional measurements of impaired visual function equate with visual disability?[END_REF] Functional ability and quality of life of these people is improved by low vision intervention. [START_REF] Stelmack | Quality of life of low-vision patients and outcomes of lowvision rehabilitation[END_REF][START_REF] Haymes | Preliminary investigation of the responsiveness of the Melbourne Low Vision ADL index to low-vision rehabilitation[END_REF] Optimal low vision care is thought to involve multiple visits with a multidisciplinary team of clinicians and rehabilitation workers, [START_REF]Recommendations for future service delivery in the UK[END_REF][START_REF] Ryan | Our better vision: What people need from low vision services in the UK[END_REF] although this is reasonably expensive. [START_REF] Gieser | Costs and utilization of end-stage glaucoma patients receiving visual rehabilitation care: A US multi-site retrospective study[END_REF] A single consultation with a low vision optometrist is the norm in the UK hospital service. [START_REF] Crossland | Thirty years in an urban low vision clinic: Changes in prescribing habits of low vision practitioners[END_REF][START_REF] Ryan | Fragmented Vision -Survey of low vision services in the UK[END_REF][START_REF] Russell | Randomised controlled trial of an integrated versus an optometric low vision rehabilitation service for patients with agerelated macular degeneration: study design and methodology[END_REF] This is usually an hour long appointment comprising history taking, refraction, advice on lighting, prescription of optical magnifiers which are loaned to patients, discussion and demonstration of electronic magnifiers and referral to other agencies as needed with no further training in the use of low vision devices.

Anecdotal reports have suggested that a follow up visit by a trained Low Vision Support Worker might be beneficial. However, Reeves and colleagues found no additional improvement with enhanced low vision intervention in patients with neovascular age-related macular degeneration. This study was a three-arm randomised controlled trial: (1) conventional hospital-based low vision care, (2) the same hospital -based low vision care followed by three home visits from a trained rehabilitation officer and (3) the hospital -based low vision care followed by home visits from a community care worker which was not vision specific. In their study, the quality of life was not improved in any group. [START_REF] Reeves | Enhanced low vision rehabilitation for people with age related macular degeneration: a randomised controlled trial[END_REF] They also assessed task performance at twelve months. For example they were asked to identify the use-by-date on supermarket grocery items.

A more comprehensive measure of functional ability related to vision is the Activity Inventory developed by Massof. A detailed description of the Massof Activity Inventory (MAI) has been published elsewhere. [START_REF] Massof | Visual disability variables. I: The importance and difficulty of activity goals for a sample of low-vision patients[END_REF][START_REF] Massof | Visual disability variables. II: The difficulty of tasks for a sample of low-vision patients[END_REF] The MAI has a hierarchical structure and is based on the Activity Breakdown Structure (ABS).

The ABS views specific activities (e.g. cooking) as goals. Each goal has a collection of tasks ( e.g. reading a recipe). For this study we used a version of the MAI that assesses difficulty achieving 50 activity goals in three domains: daily living, social, and recreational.

A further aspect of low vision is the ability to handle and use low vision devices.

We assessed this with the Manchester Low Vision Questionnaire, part 1 (MLVQ) which asks how often the low vision devices were used, the average length of time they were used, the longest time used how difficult they were to use and what kinds of difficulties were encountered.

Following the work of Reeves and colleagues, we aimed to assess whether an additional follow up by a trained support worker, a relatively low cost intervention, would be beneficial in a real life setting including all new visitors to the low vision clinic regardless of condition. In a randomised masked controlled trial we compared the effect of conventional low vision assessment with or without a second follow up on daily activities and handling of low vision aids. We extend the work of Reeves and co-workers by using the MAI questionnaire, and also assess low vision device handling using the MLVQ part 1.

Methods

This study was approved by the Local Research Ethical Committee, registered with the ISRCTN (05434212) and conformed to the CONSORT guidelines.

Patient records for those attending for a low vision assessment at the St George's low vision clinic of Moorfields Eye Hospital were assessed. Inclusion criteria were all first time attendees to the low vision clinic over the age of 18.They were written to in large print inviting them to take part in the study .

When written consent was received, the research optometrist telephoned the participants and checked they met the study inclusion criteria .Specific exclusion criteria assessed at this time included patients who were not independent, those who were not fluent in English, those who were hospital in-patients, those who previously attended a low vision assessment elsewhere, and those with serious hearing impairment. The baseline modified Massof Activity Inventory (MAI) was then administrated by telephone.

We modified the MAI by removing activities which were not relevant to the majority of British participants with low vision: driving, shooting, and leatherwork.

We were informed that this would not affect the validity of the instrument (Personal communication, Dr R Massof, 2005).

All participants received a standard low vision assessment (LVA) within 2 weeks. At this stage, patients were excluded from the study if their visual acuity was better than 6/12 and they had no significant field loss in their better seeing eye.

After the LVA visit, the participants were randomised by the low vision optometrist to the intervention and the control groups. A sealed envelope technique was used. All investigators were masked to the randomisation.

Those randomised to the intervention group had a one hour appointment with a Low Vision Support Worker (LVSW) two weeks after the initial LVA. The LVSW was a qualified dispensing optician who trained alongside Moorfields Eye Hospital trainee optometrists in basic elements of low vision assessment. He sat in with senior staff in the LVA clinic and gained experience of a multidisciplinary low vision service at The Royal National Institute of Blind People in London.

During the appointment he reviewed handling of low vision devices, discussed specific problems noted at home and if necessary issued new devices or exchanged them for something more appropriate. He also ensured that participants were aware of all services available to them through local social services and the voluntary sector. After the visit, he remained as a named contact for any further advice, given by telephone.

Those randomised to the control group had a well person check with a nurse who measured height, weight, vision and blood pressure. If they expressed difficulty with their devices they were referred back to the LVA clinic after the questionnaires were completed.

The extra appointment was approximately equal in length for those in the control and intervention group.

Approximately two weeks after the follow up visit, the research optometrist administered the MLVQ part 1 and a follow up MAI by telephone. Three months after the initial visit, the MAI and MLVQ part 1 were repeated by telephone. Data were analysed on an intention to treat basis. The principal outcome measure was task difficulty. A sample size calculation was performed using the programme PS (V3.0). [START_REF] Dupont | Power and sample size calculations. A review and computer program[END_REF] In order to detect a clinically meaningful difference between groups of 0.7 logits (corresponding to functional change of 5 lines on an ETDRS acuity chart) the required sample size is 110, or 55 per group (based on a two-tailed t-test for independent samples with a standard deviation of 1.3 logits [R Massof, personal communication]; alpha =.05, power = .8). To detect a change of 0.7 logits from before the LVA to after LVA would require only 29 patients per group (based on a two-tailed t-test for dependent samples).

Results

The records of 549 patients attending the LVA were reviewed for eligibility. 343 patients were invited to take part, and 171 patients were enrolled. The main reasons for not being enrolled were either that they had previously received low vision care elsewhere (177 participants); met the exclusion criteria (58) or declined to participate (100).

In addition, 19 patients failed to attend the LVA and 32 patients attended the LVA but decided not to continue. A total of 120 participants were randomised. Of these, 24 participants failed to complete the study, 12 from each group. 17 participants failed to attend the follow up appointment, 3 participants did not have the second interview, and 4 participants did not have the third interview. The data of the 96 participants who completed the study was analysed. Figure 1 summarises the number of patients in different stages of the study.

Effect of low vision intervention

Task ability improved from a mean value of 0.18 logit to 0.82 logit in all subjects after receiving the low vision intervention (p<0.0001). This effect is comparable to a +0.55 logMAR improvement in visual acuity. [START_REF] Dupont | Power and sample size calculations. A review and computer program[END_REF] Although the intervention group had a slightly greater increase in score, there was no significant difference in the magnitude of this improvement between the intervention and control groups: improvement was 0.73 logit for the intervention and 0.53 logit for the control group (p=0.34).

Table 2 shows the mean person measures, by group, pre-and post-intervention.

It can be seen that a significant improvement in person score occurred for both the intervention and the control groups. At three months after the intervention, the mean person measure was 0.71 for the intervention group and 0.81 for the control group. This was not significantly different from the score immediately following the intervention (matched pairs: intervention group p=0.28; control group p=0.98). This indicates that the improvement conferred by a low vision assessment does not reduce significantly over a three month period.

Intervention

Although those allocated to the intervention group had fewer problems with device use on some of the dimensions of the MLVQ: for example, 15% of the intervention group reported difficulty with device handling (control group: 25%);

2.4% reported posture problems (control group: 5.5%); and 7.3% difficulty with reading along a line (control: 16%) the differences between the two groups were not statistically significant, perhaps because of the relatively rare reporting of problems (Fishers exact test: Device handling, p=0.22; Posture problems, p=0.23; Line difficulty: p=0.63).

Discussion

Our study showed that a visit to a hospital based low vision service is related to an improvement in functional ability, as assessed using the MAI. The improvement for those seen by a conventional hospital low vision service is equivalent to an intervention which improves visual acuity by 0.55 logMAR units.

The addition of a second visit with a low vision support worker who added training in the use of devices did not improve functional ability further.

Our results differed from those of Reeves who demonstrated no improvement in quality of life following LVA. There are important differences between the two studies that may account for the apparent discrepancy. First, the follow-up period in the Reeves study was longer (twelve months as opposed to our three months), and the lack of a sustained effect at twelve months following low vision intervention has been reported elsewhere. [START_REF] Stelmack | Short-and longterm effects of an intensive inpatient vision rehabilitation program[END_REF] Second, all of their participants had age-related macular disease (AMD) whereas over half of participants in the present study had a diagnosis other than AMD. Disease progression and therefore progressive vision loss which would impact negatively on quality of life may be more dramatic for AMD. Finally, we used a more comprehensive instrument (the Massof Activity Inventory) which may be more sensitive to changes in functional ability Although our service is not multidisciplinary involving multiple visits, eccentric viewing training and the provision of electronic magnifiers, we have shown that it has a beneficial affect on quality of life. This may be further improved with an enhanced service using low vision trainers and more extensive follow up, following the model of, for example, the Veterans Administration. [START_REF] Stelmack | Outcomes of the Veterans Affairs Low Vision Intervention Trial (LOVIT)[END_REF] We have not attempted with this study to determine the utility of eccentric viewing training or more elaborate rehabilitative techniques.

Our service tends to prescribe more hand and stand magnifiers and fewer spectacle mounted telescopes or more advanced devices than other low vision clinics. [START_REF] Crossland | Thirty years in an urban low vision clinic: Changes in prescribing habits of low vision practitioners[END_REF][START_REF] Scott | Quality of life of low-vision patients and the impact of low-vision services[END_REF][START_REF] Van Rens | Results obtained with low vision aids. A retrospective study[END_REF][START_REF] Leat | The Experience of a University-Based Low Vision Clinic[END_REF] We have speculated previously that this may be as a direct consequence of not having device training available. [START_REF] Crossland | Thirty years in an urban low vision clinic: Changes in prescribing habits of low vision practitioners[END_REF] If our clinic issued more complex devices, device handling training may have been more beneficial.

Some of the limitations of our study were that we were reliant on self report of difficulties which are not always reliable and consistent. A limitation imposed by the ethics committee is that only participants who were able to see and return their consent form were able to be recruited into this study. This may have excluded those with the most severe visual impairment. The exclusion of non-English speakers may also have introduced a bias. However, this would be unlikely to contribute to the lack of difference between the control and intervention groups we report. A second limitation is that while we exceeded our recruitment goal of 110 patients, only 96 patients completed the study. However,

  This negative finding might be due to either the low sensitivity to change of the questionnaire, or disease progression in the patients assessed. The Reeves study used four questionnaires: the UK Short-Form 36 Health Survey Questionnaire, The Nottingham Adjustment Scale, The Vision Related Quality of Life Questionnaire and The Manchester Low Vision Questionnaire.

  our original power calculation assumed a standard deviation of MAI scores of 1.3 logits. The actual standard deviation obtained in the study was 1.0 logits which gives us a power of 0.92 to detect a clinically meaningful difference of 0.7 logits between groups.2 Both our intervention and control groups attended a conventional low vision assessment followed by a second visit, to either device training or a control group, hence the utility of a single LVA appointment with no follow-up visit can not directly be assessed. However we would argue that the additional visit to a well-person check should not improve vision related visual function more than a single visit to a low vision clinic. Our work therefore supports the claim that there is a clear improvement in functional ability conferred by a single visit to an optometrist led hospital based low vision clinic. An extra visit to review device handling does not further enhance the utility of this form of low vision intervention.

Acknowledgements

This study was generously funded by the Special Trustees of Moorfields Eye Hospital. The authors wish to thank Dan Ehrlich, Christine Real, Mitch Reuben, Katy Barnard and Emma Farrow for assistance with this study.


Rémi Gribonval
email: remi.gribonval@inria.fr

Keywords: degenerate blind source separation, piecewise linear separation, sparse decomposition, nonlinear approximation, Best Basis, Matching Pursuit, denoising, Wiener filter, masking, clustering

We propose a new framework, called piecewise linear separation, for blind source separation of possibly degenerate mixtures, including the extreme case of a single mixture of several sources. Its basic principle is to : 1/ decompose the observations into "components" using some sparse decomposition/nonlinear approximation technique; 2/ perform separation on each component using a "local" separation matrix. It covers many recently proposed techniques for degenerate BSS, as well as several new algorithms that we propose. We discuss two particular methods of multichannel decompositions based on the Best Basis and Matching Pursuit algorithms, as well as several methods to compute the local separation matrices (assuming the mixing matrix is known). Numerical experiments are used to compare the performance of various combinations of the decomposition and local separation methods. On the dataset used for the experiments, it seems that BB with either cosine packets of wavelet packets (Beylkin, Vaidyanathan, Battle3 or Battle 5 filter) are the best choices in terms of overall performance because they introduce a relatively low level of artefacts in the estimation of the sources; MP introduces slightly more artefacts, but can improve the rejection of the unwanted sources.

INTRODUCTION

Source separation is a problem that arises when one or several sensor(s) record data to which can contribute several generating physical processes. Perhaps the most striking example of BSS problem consists in recovering the contributions of several musical instruments to a stereophonic audio recording. If we denote by the signal emitted by the -th instrument (

) and the data recorded on the -th channel of the recording (here ), we can make the (simplistic) instantaneous linear mixture model and try to recover the source signals from the two mixtures . More generally, Blind Source Separation consists in recovering unknown sources from instantaneous mixtures . The instantaneous linear mixture model is conveniently expressed using the matrix notation [START_REF] Cardoso | Blind signal separation: statistical principles[END_REF] or where is additive noise. Note that as a general notation in this paper, we will use bold letters to denote variables that are "multichannel", such as or the mixing matrix , and plain letters to denote variables that correspond to only one channel, such as . Note also that we consider real or complex data (signals and matrices).

Considering the case of discrete signals of samples (

), with the assumption that there is no noise ( ), BSS can be seen as a factorization problem : the matrix should be factored into the matrix and the matrix . This is obviously an ill-posed problem, and its solution cannot be defined without additional assumptions, on the sources (such as independence [START_REF] Cardoso | Blind signal separation: statistical principles[END_REF] and positivity [START_REF] Oja | Blind separation of positive sources using non-negative PCA[END_REF]) or on the mixing matrix.

The most widely studied BSS situation is the (over)determined case where there is at least as many mixtures as there are sources, i.e.

. In this case, estimating the mixing matrix is sufficient to get an estimate of the sources, and the standard methods (see [START_REF] Cardoso | Blind signal separation: statistical principles[END_REF] and the references therein) have essentially the following structure : an estimate of the mixing matrix is obtained (by optimizing some contrast function which is generally highly nonlinear, or by joint diagonalization of higher order cumulants; the (pseudo)inverse of the mixing matrix is applied to the mixtures to estimate the sources as , i.e.

If there is no noise ( ) and a perfect estimate of is available, these methods provide perfect recovery of the sources. In general, there are however intrinsic limitations [START_REF] Cardoso | On the performance of orthogonal source separation algorithms[END_REF] to the accuracy of the estimation of .

In this paper we are particularly interested in the degenerate case . In this case, even if it is still possible [START_REF] Taleb | On underdetermined source separation[END_REF] to estimate the mixing matrix, the knowledge of is not sufficient to estimate the sources, because (as noted by [START_REF] Bermond | Méthodes de séparation de sources dans le cas sous-déterminé[END_REF]) the equation has an affine set of solutions. To select a prefered solution in this set, one can still choose to rely on a demixing matrix , i.e.

. However the performance of such a linear separation in the degenerate case has intrinsic limitations [START_REF] Gribonval | Proposals for performance measurement in source separation[END_REF] : "good" estimators of unknown sources from degenerate mixtures are necessarily nonlinear.

Recently, several algorithms for the separation of more sources than mixtures have been proposed [START_REF] Van Hulle | Clustering approach to square and non-square blind source separation[END_REF][START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF][START_REF] Zibulevsky | Blind source separation by sparse decomposition in a signal dictionary[END_REF][START_REF] Kisilev | Multiresolution framework for blind source separation[END_REF][START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF]. They rely on some joint representation of at least mixtures in some signal dictionary, followed by a clustering technique which is often used to build binary masks. Similarly, in the interesting (but somehow extreme) case where only one mixture is available, Roweis [START_REF] Roweis | One microphone source separation[END_REF] proposed to build binary time-frequency masks based on Hidden Markov Models (HMM) of the sources. Benaroya [START_REF] Benaroya | Wiener-based source separation with HMM/GMM using a single sensor[END_REF][START_REF] Benaroya | Non negative sparse representation for wiener based source separation with a single sensor[END_REF][START_REF] Benaroya | S éparation de plusieurs sources sonores avec un seul microphone[END_REF] combined HMM with Gaussian Mixture Models (GMM) of the sources to replace the binary masks by "adaptive Wiener filters", and Jang et al. replaced binary masks with weights based on generalized Gaussian models of the sources [START_REF] Jang | Single channel signal separation using maximum likelihood subspace projections[END_REF].

We propose a general framework, called piecewise linear separation, for the separation of possibly more sources than sensors. It generalizes the techniques for degenerate BSS that we have mentioned above. In Section 2 we introduce our framework which is based on two elements : the choice of a decomposition of the observed mixtures into components (based, e.g., on classical time-frequency or time-scale methods such as frame decompositions, Best Basis, Matching Pursuit or Basis Pursuit); the choice of local separation matrices to estimate the components of the sources from those of the mixtures. In Section 3 we discuss the possible multichannel decompositions that will serve as the first element of piecewise linear separation, and we detail two methods based on the Best Basis and Matching Pursuit algorithms. In Section 4 we discuss the computation of the local separation matrices, assuming the mixing matrix is known, based on some Bayesian priors for the sources. In Section 5 we perform numerical experiments to compare the performance of some combinations of decomposition methods and local separation strategies. We conclude by discussing what seem to be the main challenges as well as the most promising research directions to design better algorithms within the piecewise linear separation framework.

PIECEWISE LINEAR SEPARATION

Several authors have proposed BSS algorithms for the possibly degenerate case that are based on some time-frequency/timescale representation of the data followed by binary masking [START_REF] Van Hulle | Clustering approach to square and non-square blind source separation[END_REF][START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF][START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF][START_REF] Roweis | One microphone source separation[END_REF]. The underlying model is that at most one source is "active" in each component of the representation. In the one microphone setting, some authors [START_REF] Benaroya | Wiener-based source separation with HMM/GMM using a single sensor[END_REF][START_REF] Benaroya | Non negative sparse representation for wiener based source separation with a single sensor[END_REF][START_REF] Benaroya | S éparation de plusieurs sources sonores avec un seul microphone[END_REF][START_REF] Jang | Single channel signal separation using maximum likelihood subspace projections[END_REF] proposed to replace binary masking with some smoother form of filtering, where each component is split adaptively to obtain the components of the sources. Here we want to gather the main ideas that have emerged and propose a global framework for degenerate BSS : piecewise linear separation. The key observation is that a good data representation often makes it possible to decompose a single degenerate BSS problem into several (over)determined problems. The basic principle is simply to :

1. decompose the observations into "components" ; 2. perform separation on each component using a "local" separation matrix.

Before we discuss how to perform these two steps, let us see how piecewise linear separation works in an ideal case.

Ideal model; some notations

Let us assume the sources can be expressed as a sum , which may be written componentwise , where we will call the -th "component" of the source . For example, if is an orthonormal basis we have such a decomposition with . More general decompositions will be considered in Section 3. Assume in addition that for each component, the set of "active" sources is known and contains at most entries. From the noiseless mixing model , we easily derive a decomposition of the observations, with where is the card matrix with columns , ( is the -th column of the mixing matrix ), and is the card matrix with rows , . Thus, there exist a split of the original problem into an equivalent collection of subproblems :

Because card , each subproblem is (over)determined, so we get

(2)

(3)

The ideal case requires a noiseless problem with several assumptions that are generally unrealistic. We are indeed supposed to know : the mixing matrix ;

how to decompose so as to "match" the decomposition of ; the set of "active" sources on each component, which should satisfy card ;

Under these ideal assumptions, we have seen that it is possible to perfectly recover the sources.

Principle of piecewise linear separation algorithms

Of course, real BSS problems may include additive noise and, most of all, they are blind :

the mixing matrix is unknown;

it is not known in advance which (and how many) sources are "active" in which component; it may not be possible to claim that only one source is active in each component;

in the extreme -but interestingone microphone setting, one may even have to relax the assumption that at most sources are active on one component.

Consider for example the separation of instruments in a commercial musical recording. Not only are the sources dependent (at least at a high level point of view : they follow the same musical score), they are also temporally synchronized and have common harmonics, so there almost certainly exist time-frequency components that contain contributions of several sources. If one ever wants to address audio BSS problems of this nature (and it seems to be the implicit dream of almost every researcher in the field of BSS!), the assumption that at most one source contributes to each component is not quite realistic.

We propose the following general algorithmic structure for piecewise linear BSS, where some of the steps may be performed jointly : In Section 3 we discuss the decomposition step (step 1), which may rely on nonlinear approximation and sparse decomposition techniques. The local separation step (step 2) is probably the most critical one. In Section 4 we discuss how it can be performed based on Bayesian priors when one knows an estimate of the mixing matrix . Estimating is a problem by itself which can be adressed by various techniques [START_REF] Taleb | On underdetermined source separation[END_REF][START_REF] Van Hulle | Clustering approach to square and non-square blind source separation[END_REF][START_REF] Theis | A histogram-based overcomplete ICA algorithm[END_REF][START_REF] Waheed | Algebraic overcomplete independent component analysis[END_REF][START_REF] Lathauwer | Independent component analysis of largely underdetermined mixtures[END_REF][START_REF] Albera | Sixth order blind identification of underdetermined mixtures (BIRTH) of sources[END_REF] but its not dealt with in this paper.

Before we discuss both steps and show that that this general framework covers many recently proposed algorithms, let us emphasize that it also makes it possible to design new algorithms that : deal with more than one "active" source per component (and sometimes even more than sources, such as in the one microphone setting); decide globally which sources are active in which component, by taking into account the dependencies between components (harmonic relations, persistence of instantaneous frequency across time, persistence of transients across scale, . . . ) rather than making independent decisions for each component.

MULTICHANNEL SIGNAL DECOMPOSITIONS

Besides the raw (time-domain) representation of a signal, there are many ways to decompose a (monochannel) signal : linear time-frequency/time-scale transforms such as wavelets bases, wavelet frames, Gabor frames, Wilson bases [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]; adaptive methods using local cosine bases, wavelet packets, with the Best Basis algorithm [START_REF] Coifman | Entropy-based algorithms for best basis selection[END_REF]; dictionary decompositions using Matching Pursuit [START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF][START_REF] Mallat | Matching pursuit with time-frequency dictionaries[END_REF] or Basis Pursuit [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF]. In this section we discuss how these monochannel signal decompositions can be adapted to get multichannel ones.

Notations

First, let us introduce a few notations. For multichannel signals ( denotes the real transpose while denotes the Hermitian transpose) and with an arbitrary number of channels we define the inner product and its associated norm (5) [START_REF] Gribonval | Proposals for performance measurement in source separation[END_REF] where denotes complex conjugation.

Frame decompositions

Frames are families of "atoms" such that for all (monochannel) signals where and . For any frame, there exists a dual frame such that every signal has the frame decomposition For multichannel signals (of any dimension ) we define, (by abuse of notation we use the same notation for all dimensions ) and get the desired multichannel decomposition With such a decomposition, solving the initial BSS problem ( 1) is equivalent to solving the collection of "sub-problems"

(notice that and have rows while has rows).

The most basic frame decomposition is the trivial representation where the atoms are simply Diracs , it is used in Van Hulle's clustering approach to degenerate BSS [START_REF] Van Hulle | Clustering approach to square and non-square blind source separation[END_REF]. Another classical transform is the Short Time Fourier Transform (STFT) used in the Degenerate Unmixing and Estimation Technique (DUET algorithm) of Jourjine et al. [START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF].

To the STFT correspond time-frequency atoms localized at time and frequency (see, e.g., the book of Mallat [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]). All sorts of Gabor and wavelet frames as well as local cosine bases or wavelet packets bases can be used similarly.

Other linear decompositions

Instead of decomposing the signals along "atoms", it is also possible to cut them linearly into larger pieces (or "molecules") that correspond to higher dimensional subspaces of the signal space. For example, in overlap-add methods (OLA), it is classical to decompose a signal into windowed pieces where the shifted windows sum to one. Similarly, it is possible to decompose a signal into subbands using a family of filters. For example, based on an orthonormal basis of wavelets [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] one can build :

Which linear decomposition is better for piecewise linear separation certainly depends on the nature of the sources, and it requires numerical experiments on large databases to choose a linear representation that is well adapted to a target application. For instance, DUET [START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF] is based on the STFT on the ground that independent audio sources seem to be quasi "W-disjoint orthogonal" [START_REF] Rickard | On the approximate w-disjoint orthogonality of speech[END_REF][START_REF] Balan | Statistical properties of STFT ratios for two channel systems and applications to blind source separation[END_REF], while Jang et al. [START_REF] Jang | Single channel signal separation using maximum likelihood subspace projections[END_REF] rely on sparse coding [START_REF] Field | Emergence of simple-cell receptive field properties by learning a sparse code for natural images[END_REF][START_REF] Bell | The 'independent components' of natural scenes are edge filters[END_REF][START_REF] Bell | Learning the higher-order structure of a natural sound[END_REF][START_REF] Lewicki | Efficient coding of natural sounds[END_REF][START_REF] Abdallah | If edges are the independent components of natural images, what are the independent components of natural sounds?[END_REF] to compute the representation used for their algorithm. As an alternative to selecting a priori the linear transform based on knowledge of the class of sources we want to separate, it is possible to choose it adaptively depending on the observed mixtures , using nonlinear optimization techniques such as Best Basis [START_REF] Coifman | Entropy-based algorithms for best basis selection[END_REF] (BB) or Matching Pursuit [START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF][START_REF] Mallat | Matching pursuit with time-frequency dictionaries[END_REF] (MP).

Best orthogonal basis

Assume we have at hand a "library" of orthonormal bases and some cost function that predicts the performance of a basis for a target application. The principle of Best Basis (BB) is simply to pick up the best basis in the library according to the cost function. When the target application is the nonlinear approximation of a monochannel signal , a typical cost function will be, for :

where is some concave function [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]. Such cost function will be small when the energy of is well concentrated on a few coefficients, and it will take the largest value when the energy is spread equally on all coefficients. For nice treestructured libraries such as wavelet packets or local cosines, the best basis can be found with a fast search, provided that the cost function is additive [START_REF] Coifman | Entropy-based algorithms for best basis selection[END_REF]. Once the basis is selected, we can perform the decomposition using the linear transform corresponding to , i.e.

In the case of multichannel signals, assume we have at hand some cost function . We can adopt the BB strategy by selecting and decomposing with

The most straightforward choice of a cost function is certainly where measures the joint energy of the channels, and each term gives the relative amount of joint energy that is carried by the -th component.

Basis Pursuit

Zibulevsky and Pearlmutter [START_REF] Zibulevsky | Blind source separation by sparse decomposition in a signal dictionary[END_REF] proposed BSS algorithms based on sparse decompositions of the sources in some redundant signal dictionary of atoms . The dictionary typically consists in a family of time-frequency atoms , or wavelets , etc, but other less standard dictionaries can be considered thay may be estimated from training data [START_REF] Field | Emergence of simple-cell receptive field properties by learning a sparse code for natural images[END_REF][START_REF] Bell | The 'independent components' of natural scenes are edge filters[END_REF][START_REF] Bell | Learning the higher-order structure of a natural sound[END_REF][START_REF] Lewicki | Efficient coding of natural sounds[END_REF][START_REF] Abdallah | If edges are the independent components of natural images, what are the independent components of natural sounds?[END_REF][START_REF] Kreutz-Delgado | Convex/schur-convex (csc) log-priors and sparse coding[END_REF] or may combine several classical dictionaries into a larger one [START_REF] Mallat | Matching pursuit with time-frequency dictionaries[END_REF][START_REF] Benaroya | Représentations parcimonieuses pour la séparation de sources avec un seul capteur[END_REF][START_REF] Starck | Image decomposition : Separation of textures from piecewise smooth content[END_REF][START_REF] Daudet | Représentations structurelles de signaux audiophoniques : m éthodes hybrides pour des applications à la compression[END_REF]. Because of the assumed redundancy of the dictionary, any monochannel signal has infinitely many representations with coefficients

, so one has some freedom in choosing the most convenient set of coefficients. Zibulevsky and Pearlmutter choose a "sparse" decomposition among all the possibilities according to some "sparseness" criterion, which is derived in [START_REF] Zibulevsky | Blind source separation by sparse decomposition in a signal dictionary[END_REF] from a probabilistic model of the unknown coefficients. When the sparseness measure is the norm (it corresponds to a Laplacian model on the unknown coefficients), the coefficients are recovered using Basis Pursuit (BP) [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF]. They are generally computed with Linear Programming algorithms which are computationally intensive. In the multichannel case, Zibulevsky and Pearlmutter [START_REF] Zibulevsky | Blind source separation by sparse decomposition in a signal dictionary[END_REF] propose a joint estimation of the sources and the matrix via a MAP optimization under the Laplacian model.

Matching Pursuit

BP is computationally intensive and tricky to implement with arbitrary dictionaries, even in the monochannel case. To the opposite, the Matching Pursuit algorithm is quite generic and easy to implement, as well as easily generalized to multichannel decompositions. Moreover, it shares some of the good properties of Basis Pursuit for the perfect recovery of very sparse expansions in well-behaved dictionaries [START_REF] Gilbert | Approximation of functions over redundant dictionaries using coherence[END_REF][START_REF] Tropp | Greed is good : Algorithmic results for sparse approximation[END_REF][START_REF] Gribonval | Approximation with highly redundant dictionaries[END_REF][START_REF] Gribonval | Exponential convergence of Matching Pursuit in quasi-incoherent dictionaries[END_REF]. Matching Pursuit (MP) [START_REF] Mallat | Matching pursuit with time-frequency dictionaries[END_REF] is a strategy that selects a sequence of indexes of atoms, and iteratively computes some residuals (starting from ) as [START_REF] Van Hulle | Clustering approach to square and non-square blind source separation[END_REF] In MP the sequence is iteratively selected depending on the signal :

Hence, after iterations, is decomposed as

The author proposed a version of the Matching Pursuit for stereophonic audio signals [START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF] which we readily extend to an arbitrary number of channels. After iterations, it provides a decomposition of with

(9) (10)
As a natural extension of the monochannel case, we propose to select the indexes as [START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF] so as to pick up at each iteration a component that carries as much as possible of the joint energy of the multichannel residual. Such a multichannel MP shares the same convergence properties as the monochannel one [START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF][START_REF] Leviatan | Simultaneous approximation by greedy algorithms[END_REF], that is to say .

FROM BINARY MASKING TO LOCAL LINEAR SEPARATION

The first step in piecewise linear separation is a decomposition step, and we have seen several possible choices to perform it. In this section we discuss the second step, that is to say the computation of local separation matrices . As mentioned in the introduction, there are methods to estimate the mixing matrix even in the degenerate case [START_REF] Taleb | On underdetermined source separation[END_REF][START_REF] Van Hulle | Clustering approach to square and non-square blind source separation[END_REF][START_REF] Theis | A histogram-based overcomplete ICA algorithm[END_REF][START_REF] Waheed | Algebraic overcomplete independent component analysis[END_REF][START_REF] Lathauwer | Independent component analysis of largely underdetermined mixtures[END_REF][START_REF] Albera | Sixth order blind identification of underdetermined mixtures (BIRTH) of sources[END_REF]. However, in the degenerate case, the knowledge of the mixing matrix is not sufficient to recover the sources [START_REF] Bermond | Méthodes de séparation de sources dans le cas sous-déterminé[END_REF]. In this section, we want to concentrate on source recovery, so we assume that an estimate of the mixing matrix is available. Choosing a particular solution in the affine set of solutions to the equation requires prior models of the sources, and the selection can rely on Bayesian estimators. In this section, we model the noise as Gaussian, spatially and temporally white, with sample variance . Then, the Maximum A Posteriori (MAP) estimator is MAP [START_REF] Roweis | One microphone source separation[END_REF] where , while the Conditional Mean (CM), which minimizes the mean square error, is

Because denoising is certainly the most classical source separation problem, we start this section by showing how two classical denoising techniques, hard-thresholding and Wiener filtering, fit in the piecewise linear separation framework.

In the remainder of the section we propose several prior models of the sources, which lead to various choices of local separation matrices. We recover as special cases some known techniques such as binary masking [START_REF] Van Hulle | Clustering approach to square and non-square blind source separation[END_REF][START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF][START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF][START_REF] Roweis | One microphone source separation[END_REF], separation by sparse decompositions [START_REF] Zibulevsky | Blind source separation by sparse decomposition in a signal dictionary[END_REF][START_REF] Kisilev | Multiresolution framework for blind source separation[END_REF] and adaptive Wiener filtering [START_REF] Benaroya | Wiener-based source separation with HMM/GMM using a single sensor[END_REF][START_REF] Benaroya | Non negative sparse representation for wiener based source separation with a single sensor[END_REF][START_REF] Benaroya | S éparation de plusieurs sources sonores avec un seul microphone[END_REF], but also get new possibilities. In Section 5 we will compare the performance of combinations of some of the decomposition methods proposed in Section 3 with some of the local separation strategies introduced below.

Denoising

We consider an observed signal ( is the source of interest, is the noise) and the corresponding mixing matrix is . In the hard-thresholding strategy the source of interest is estimated by : 1/ computing the components where is an orthonormal basis (e.g., an orthonormal wavelet basis [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]); 2/ deciding whether is active in by testing whether where is a threshold, and applying the local separation matrices if is considered active, if it is not. Wiener filtering can also be seen as a piecewise linear separation algorithm : the components correspond to the Fourier decomposition with and we assume all sources are simultaneously active with known variance . Local separation matrices are expressed as

Binary masking

The masking approaches to degenerate BSS [START_REF] Van Hulle | Clustering approach to square and non-square blind source separation[END_REF][START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF][START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF][START_REF] Roweis | One microphone source separation[END_REF] estimate from a piece of the observations by : 1/ estimating the index of the only "active" source in the -th component; 2/ recovering the sources by "masking" the components of one of the observed channels : [START_REF] Benaroya | Non negative sparse representation for wiener based source separation with a single sensor[END_REF] The choice of which channel is masked is arbitrary. Binary masking can be seen as piecewise linear separation with and the decomposition is either the raw data [START_REF] Van Hulle | Clustering approach to square and non-square blind source separation[END_REF], a STFT [START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF][START_REF] Roweis | One microphone source separation[END_REF] or adaptive time-frequency representations [START_REF] Zibulevsky | Blind source separation by sparse decomposition in a signal dictionary[END_REF][START_REF] Kisilev | Multiresolution framework for blind source separation[END_REF][START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF].

When only one mixture is available, Roweis [START_REF] Roweis | One microphone source separation[END_REF] proposed to estimate the masks using Hidden Markov Models (HMM); when at least two mixtures are available, they can be estimated by exploiting the so called spatial diversity through clustering [START_REF] Van Hulle | Clustering approach to square and non-square blind source separation[END_REF][START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF][START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF]. Spatial diversity can still be exploited when more than one source may be active in each component [START_REF] Kisilev | Multiresolution framework for blind source separation[END_REF], provided that card where is called the hidden activity state of the sources on the -th component. Below we propose different strategies to estimate the hidden activity state depending on the choice of a prior model .

Disjoint orthogonal model.

The most common assumption on the hidden activity states is that card , i.e. exactly one source is active in each component (in DUET [START_REF] Jourjine | Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures[END_REF], this is called "W-disjoint orthogonality"). This corresponds to a prior probability . Based on this model, for each hypothesis , Eqs. ( 2)-( 3)) give the Maximum Likelihood (ML) estimate of the "would be active" source where is the -th column of the mixing matrix (remember we assume is known). Thus, the MAP estimate of the hidden activity state is with [START_REF] Benaroya | S éparation de plusieurs sources sonores avec un seul microphone[END_REF] Eventually the sources are estimated piecewise linearly as [START_REF] Jang | Single channel signal separation using maximum likelihood subspace projections[END_REF] i.e. with local separation matrices [START_REF] Theis | A histogram-based overcomplete ICA algorithm[END_REF] Note that, to the opposite of Eq. ( 14), there is no need to arbitrarily choose the channel that is masked : the components of the sources are estimated by projecting orthogonally the components of the mixtures onto the direction of the corresponding column of the mixing matrix. We will compare numerically in Section 5 the performance of separation based on Eq. ( 14) and Eq. ( 16).

Other models. With any prior

on the hidden activity state such that card , for each hypothesis on , piecewise linear separation is given by Eqs. ( 2)-( 3), hence the MAP estimate of is with where is the variance of the noise on the -th component. For example, if and , then the usual choice given by Eq. ( 15) is replaced by whenever This is similar to denoising by hard-thresholding [START_REF] Donoho | Ideal denoising in an orthonormal basis chosen from a library of bases[END_REF][START_REF] Donoho | Denoising by soft-thresholding[END_REF], where small components are considered as pure noise.

In [START_REF] Zibulevsky | Blind source separation by sparse decomposition in a signal dictionary[END_REF][START_REF] Kisilev | Multiresolution framework for blind source separation[END_REF], BSS algorithms based on sparse decompositions are proposed. It is not difficult to check that these algorithms recover the sources piecewise linearly just as in Eqs. ( 2)-( 3): the hidden activity state , which satisfies card , is estimated by minimizing . The underlying model is Laplacian for the source components and uniform over all hidden activity states of size .

Masking based on structure. When only one channel is available, spatial diversity can no longer be exploited to compute the binary masks. Instead, other prior information has to be exploited. One possibility is to use global priors that may take the form of HMM [START_REF] Roweis | One microphone source separation[END_REF]; another one consists in using the fact that different sources may yield different "types" of components. For instance, on single channel audio signals, transients and sustained parts can be separated based on a Matching Pursuit decomposition with a Gabor dictionary [START_REF] Gribonval | Analysis of sound signals with high resolution matching pursuit[END_REF][START_REF]Temps-fréquence : concepts et outils, ch. Analyse temps-fréquence linéaire I : Reprsentations type Fourier[END_REF]; similarly, it is possible to separate edges from textures in images [START_REF] Starck | Image decomposition : Separation of textures from piecewise smooth content[END_REF] using the fact that each of theses "sources" has a sparse representation in a different basis [START_REF] Benaroya | Représentations parcimonieuses pour la séparation de sources avec un seul capteur[END_REF], and the basis corresponding to different sources are "incoherent". Recent results on sparse decompositions and nonlinear approximation with dictionaries [START_REF] Gilbert | Approximation of functions over redundant dictionaries using coherence[END_REF][START_REF] Tropp | Greed is good : Algorithmic results for sparse approximation[END_REF][START_REF] Gribonval | Exponential convergence of Matching Pursuit in quasi-incoherent dictionaries[END_REF][START_REF] Donoho | Uncertainty principles and ideal atomic decompositions[END_REF][START_REF] Elad | A generalized uncertainty principle and sparse representations in pairs of bases[END_REF][START_REF] Gribonval | Sparse decompositions in unions of bases[END_REF][START_REF] Donoho | Optimally sparse representation in general (non-orthogonal) dictionaries via minimization[END_REF][START_REF] Fuchs | On sparse representations in arbitrary redundant bases[END_REF] have given theoretical ground to these techniques which, again, have a piecewise linear form: after a nonlinear decomposition step, each component is used to recover the source to which it belongs.

Smooth masking

In theory, binary masking techniques can perfectly recover sources that globally linear BSS algorithms cannot recover. In practice, because the underlying model is not perfect, they often introduce artifacts due to the introduction of unnatural zeroes in the representation of the estimated sources. For audio sources, this leads to artifacts similar to the well known "musical noise" or "pipe noise" which is commonly encountered in transform-based coding.

Some authors [START_REF] Benaroya | Wiener-based source separation with HMM/GMM using a single sensor[END_REF][START_REF] Benaroya | Non negative sparse representation for wiener based source separation with a single sensor[END_REF][START_REF] Benaroya | S éparation de plusieurs sources sonores avec un seul microphone[END_REF][START_REF] Jang | Single channel signal separation using maximum likelihood subspace projections[END_REF] have proposed algorithms for single channel BSS based on "smoother" forms of masking that do not bring in the thresholding effect of binary masks. In these approaches, training data is first used to learn the parameters of generalized Gaussian models [START_REF] Jang | Single channel signal separation using maximum likelihood subspace projections[END_REF] (resp. Gaussian mixture models (GMM) [START_REF] Benaroya | Wiener-based source separation with HMM/GMM using a single sensor[END_REF][START_REF] Benaroya | Non negative sparse representation for wiener based source separation with a single sensor[END_REF][START_REF] Benaroya | S éparation de plusieurs sources sonores avec un seul microphone[END_REF]) of the two sources; separation is done piecewise linearly on new data with , where is estimated by a MAP approach [START_REF] Jang | Single channel signal separation using maximum likelihood subspace projections[END_REF] or by combining different Wiener filters [START_REF] Benaroya | Wiener-based source separation with HMM/GMM using a single sensor[END_REF] to get the conditional mean (CM) estimator (see Eq. ( 13)).

When no training data is available, one has to design other techniques that rely on a more generic prior model with few or no parameters to estimate. With this aim, we propose a "smooth masking" strategy which, by analogy with GMM, we call Masking Mixture Model. It is based on hidden activity state models just as in binary masking, but MAP estimation is replaced by CM.

Masking Mixture Models (MMM).

Assume we are given a prior on the hidden activity states such that card . For each hypothesis on , Eqs. ( 2)-( 3) provide implicitly a local separation matrix . The CM estimate of is with [START_REF] Waheed | Algebraic overcomplete independent component analysis[END_REF] where we can compute the likelihood using the fact that

In this strategy, the only parameter to adjust is the variance of the noise. In the limit where we recover the binary masking strategy, but additional research is necessary to find out good strategies to adjust .

SOME EXPERIMENTAL RESULTS

So far, we have introduced several pieces (decomposition methods and local separation strategies) that can be assembled to make BSS algorithms within our proposed global framework of piecewise linear separation. In this section we gather the results of some experiments of source separation obtained with different combinations of these pieces. The experiments were performed on a stereophonic (M=2) instantaneous mixture of three normalized sources ( cello, drums, piano, the sampling rate was and the number of samples was which corresponds to a duration of about ) with known mixing matrix. In all experiments, we have used explicitly the knowledge of to perform the local separation based on the various strategies. Estimating is a difficult problem by itself, but the goal of these experiments is to compare the decomposition methods and the local separation strategies independently of the quality of the estimate of . The dataset (original sources, mixing matrix, mixtures) of the experiment is available online [START_REF]Ressources pour la séparation de signaux audiophoniques[END_REF] and was already used in the papers [START_REF] Gribonval | Proposals for performance measurement in source separation[END_REF][START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF] 1. Comparison of the performance of some piecewise linear separation algorithms on an instantaneous stereophonic mixture of three sources. For each decomposition method, we applied the local separation strategy corresponding to Eq. ( 16) based on the true mixing matrix. We measured SDR SDR as well as the corresponding SIR and SAR figures. The performance of the best global linear separation is indicated as a reference.

Evaluation criteria

It is well known that blind source separation can only recover the sources up to gain and permutation. Letting apart the permutation problem, because of the unknown gain factor, a plain SNR does not display correctly the separation performance, and many authors instead use the SNR between and . We showed in [START_REF] Gribonval | Proposals for performance measurement in source separation[END_REF] that even this modified performance measure does not scale intuitively : in the worst case, when the estimate is orthogonal to the true source, it yields an SNR of only dB. Instead, a slight modification called the Source to Distortion Ratio (SDR) [START_REF] Gribonval | Proposals for performance measurement in source separation[END_REF] scales from dB to dB :

SDR

Moreover, in the degenerate case, efficient BSS algorithms are nonlinear because a perfect globally linear separation is impossible. In [START_REF] Gribonval | Proposals for performance measurement in source separation[END_REF] we proposed two other measures, the Source to Interference Ratio (SIR) and the Source to Artefacts Ratio (SAR) that measure respectively the amount of distortion due to remaining interferences of the unwanted sources, and the distortion due to nonlinearities in the algorithms, such as the thresholding effects. Thus, any algorithm that performs separation in a globally linear manner produces a SAR of dB. Matlab routines to compute this performance figures are available online [START_REF]Ressources pour la séparation de signaux audiophoniques[END_REF].

Experiments

We implemented the multichannel Best Basis algorithm using Wavelab 802 [START_REF] Donoho | Wavelab 802 for Matlab5[END_REF] (our source code is available online [START_REF]Best Basis Blind Source Separation (B3S2) software[END_REF]) and the multichannel MP algorithm using LastWave [START_REF] Bacry | LastWave software[END_REF]. We performed a series of experiments where we used the following decomposition methods : every cosine packets and wavelet packets bases available in Wavelab; MP with a Gaussian multiscale Gabor dictionary, with dyadic scales ranging from to . Two local separation strategies were tested : the standard one (see Eq.( 14)) where the masks are applied to one channel (the left one or the right one) which is chosen arbitrarily; the one we have derived from the disjoint orthogonal prior model (see Eq. ( 16)) where a projection on columns of the mixing matrix is used instead of an arbitrary choice. For each combination of decomposition method and local separation strategy we computed the figures SDR , SDR , SDR , SIR , SIR , SIR , SAR , SAR and SAR for each source, where and stand respectively for the left, right and projection local separation strategies. In addition, to serve as a baseline, we also computed the figures SDR , SIR , SAR corresponding to the best globally linear separation [START_REF] Gribonval | Proposals for performance measurement in source separation[END_REF].

Results

The results are summarized in Table 1. The first line of the table displays the best performance that can be expected from a globally linear separation strategy [START_REF] Gribonval | Proposals for performance measurement in source separation[END_REF], which shows that this BSS problem cannot be solved acceptably without relying on a nonlinear BSS algorithm.

We performed a first series of experiments to compare multichannel BB with the libraries of bases (local cosine packets and wavelet packets) that were available in Wavelab. A striking observation was the poor behaviour of the standard local separation strategy compared to the new one based on projection, for all BB decompositions. Over all BB decompositions and all sources, we observed that SDR SDR SDR dB SIR SIR SIR dB SAR SAR SAR dB which shows that choosing the projection strategy never degrades significatively the performance compared to the usual method. On the contrary, for every BB decomposition and every source, SDR SDR SDR (similar relations hold for SIR and SAR), hence masking the "wrong" channel in the standard strategy leads to a systematic loss of performance compared to the projection strategy. The systematic loss can be non negligible : in the case of the third source (piano) we observed that, for every BB decomposition

SDR SDR SDR dB SIR SIR SIR dB SAR SAR SAR dB
The loss of performance was as bad as dB for the first source (cello) with the SDR measure.

Given these observations, the comparison between various BB decomposition can essentially be made by combining it with the projection strategy Eq. ( 16) for local separation. The best choice (at least on this dataset) is the wavepacket library based on the Battle 5 filter. In Table 1 we display the results obtained with the five libraries that were most often ranked in the five best performing : the cosine packets, and the wavelet packets based on the Beylkin, Vaidyanathan, Battle 3 and Battle 5 filters. Compared to the best globally linear separation, the improvement in performance achieved through piecewise linear separation with BB decomposition is clear : for the Battle 5 filter wave packets, not only does the SDR increase by up to more than dB for the cello and drums sources, but most of all the SIR SIR figures are improved by at least dB for the piano and more than dB for the other sources. As indicated by the SIR figures, these nonlinear BSS algorithms achieve a good rejection of the unwanted sources, but their nonlinearity introduces artefacts, which are indicated by the SAR figures. In fact, the SDR figures show that the distortion due to artefacts completely dominates the remaining interferences of the unwanted sources, i.e. we have SDR SIR SAR SAR. We believe the MMM local separation strategy will provide a good compromise between the rejection of the unwanted sources and the nonlinear artefacts.

In addition to the experiments with BB, we made an experiment based on stereo MP [START_REF]Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture[END_REF] with atoms. For the cello and drums sources, the comparison between the local separation strategy based on the projection method and the standard one leads to the very same observations as with BB, and it seems to be a good choice to rely on Eq. ( 16). The same observation is true in terms of SIR for the piano source, hence Eq. ( 16) never noticeably degrades the rejection of unwanted sources. However, for the piano source, the new strategy degraded the SAR (and the SDR which, again, is dominated by artefacts) by about dB compared to standard binary masking of the right channel. Table 1 displays the results respectively with standard masking applied on the left channel, on the right one and with the projection strategy. 1 leads to some interestingly observations. First, these piecewise linear separation algorithms obviously outperform any globally linear separation algorithm both in terms of global separation performance and rejection of the unwanted sources, but they logically introduce nonlinear artefacts. Indeed, the artefacts are currently the limiting factor for their overall performance, and decreasing the level of artefacts should improve the SDR figure. For applications where it is important to minimize the level of artefacts, with this dataset, one would probably perform the decomposition based on BB with the Battle 5 filter (with local separation based on Eq. ( 16)). If one needs to reject the unwanted sources, at the possible price of a higher level of artefacts, then the Battle 5 choice competes with MP : even if the latter degrades the SIR by about dB for the drums source, it increases it by more than dB for the cello and piano sources.

The comparison of BB and MP in Table

CONCLUSION

In this paper, we have proposed a new global framework, called piecewise linear separation, for blind source separation of possibly degenerate mixtures. The framework is based on a combination of a decomposition of the mixtures into elementary components and local linear separation strategies on each component. We have shown that it covers many existing BSS algorithms, as well as some denoising algorithms. We proposed new multichannel decomposition methods by generalizing the Best Basis (BB) and the Matching Pursuit (MP) algorithms, and new local separation strategies besides the classical binary masking strategy. We have performed experiments with various combinations of the proposed decomposition methods and local separation strategies, and showed that a new local separation strategy generally performs better than the standard one. On the dataset used for the experiments, it seems that BB with either cosine packets of wavelet packets (Beylkin, Vaidyanathan, Battle 3 or Battle 5 filter) are the best choices in terms of overall performance because they introduce a relatively low level of artefacts in the estimation of the sources; MP introduces slightly more artefacts, but can improve the rejection of the unwanted sources.

While this paper has focussed on the issue of recovering the sources from a degenerate mixture assuming we know the mixing matrix , this assumption cannot be made in true BSS problems. We are currently investigating the possibility to estimate and the sources in a iterative way : given the current estimate , piecewise linear separation would provide estimates of the sources and a measure of likelihood of the most likely set of hidden activity states. An EM algorithm would be used to update the estimate of .

Piecewise linear separation improves the separation performance compared to globally linear separation, but its overall performance is limited by the artefacts it introduces. We believe it is possible to improve the overall performance (SDR) by reducing the level of artefacts (SAR) without degrading too much the ability to reject unwanted sources (SIR). With this aim, the use smooth forms of local separation such as Masking Mixture Models and Gaussian Mixture Models, is under investigation. Eventually, we believe it is worth building models of dependencies between the hidden activity states of different components : when instruments are mixed in a commercial musical recording, the sources are temporally synchronized (the musicians play together) and have common harmonics (they are tuned together). Moreover, transients and harmonic lines give rise to quite structured time-frequency representations [START_REF] Daudet | Représentations structurelles de signaux audiophoniques : m éthodes hybrides pour des applications à la compression[END_REF][START_REF] García | Tracking of partial for additive sound synthesis using hidden Markov models[END_REF][START_REF] Gribonval | Harmonic decomposition of audio signals with matching pursuit[END_REF]. Hidden Markov Chains [START_REF] Roweis | One microphone source separation[END_REF] and Hidden Markov Trees [START_REF] Crouse | Wavelet-based signal processing using hidden markov models[END_REF] are probably good models to investigate.

Table

  .

	Method	SDR	SDR	SDR	SIR	SIR	SIR	SAR	SAR	SAR
	best globally linear
	BB (cosine packets)
	BB (Beylkin)
	BB (Vaidyanathan)
	BB (Battle 3)
	BB (Battle 5)
	MP (2400 atoms), left masking
	MP (2400 atoms), right masking
	MP (2400 atoms), projection

Acknowledgments

Many thanks to Thomas Reulos for his precious help in the implementation of multichannel MP with LastWave [53] and in the numerical experiments. The author is also very grateful to Laurent Benaroya, Frédéric Bimbot and Guillaume Gravier for their useful suggestions, and to Emmanuel Vincent, Cédric Févotte and for their comments on an early version of this paper.


Wojciech Nogala

Anna Celabanska

Marcin Gunther Wittstock

Anna Celebanska

Marcin Opallo
email: mopallo@ichf.edu.pl

and its application in a zinc-dioxygen cell

Keywords: Oxygen Reduction Reaction, Cathode, Cyclic Voltammetry, Gas Diffusion Electrode, Hybrid Bilirubin oxidase, Biocatalysis, Electroreduction, Oxygen, Oxygen permeability, Scanning 24 electrochemical microscopy (SECM), Silicate, Sol-gel processes, Zinc, 25 26

Bioelectrocatalytic carbon ceramic gas electrode for reduction of dioxygen and its application in a zinc-dioxygen cell

Carbon ceramic electrodes (CCE) were developed in the early 1990's [1]. They consist of dispersed carbon 28 particles embedded in a silicate matrix. This hybrid material is produced by a sol-gel process followed by drying.

29

The most typical example of CCE, prepared for the use in aqueous electrolyte is obtained by sol-gel processing 30 of methyltrimethoxysilane (MTMOS) mixed with carbon microparticles [2]. Methyl groups attached to the 31 silicate network render the pores of the material hydrophobic. Consequently the electrode body is not flooded 32 when immersed into an aqueous solution. Moreover, the porous electrode material can be penetrated by 33 dioxygen up to the matrix|aqueous electrolyte interface inside the electrode body. This property was exploited by 34 the Lev group for preparation carbon ceramic gas electrodes (CCGE) exhibiting dioxygen reduction 35 electrocatalysis with the help of immobilised porphyrins or redox polymer catalyst [3][4][5][6][7]. Bioelectrocatalytic

36

CCE for the reduction of dioxygen dissolved in aqueous solutions has already been reported [8][9][10][11]. Here we 37 combine for the first time the biocatalytic dioxygen reduction with an air-breathing CCE.

38

The construction of dioxygen-permeable biocathode and its application in biofuel cells was recently proposed 39 [12][13][14][15], because dioxygen flux through porous electrode can be superior to O 2 diffusion in electrolyte and higher 40 current densities can be obtained [12]. Additionally, it allows using dioxygen-free electrolytes which improves 41 the operation of some devices. For example, in dioxygen-glucose biofuel cells [16][17][18], electron transfer to the

78

Procedures 79

The hydrophobic sol-gel matrix was prepared as already described [1]. The hydrolyzed sol was obtained by 80 mixing 500 µL of MTMOS with 750 µL of methanol. After addition of 25 µL of 11 M HCl, it was sonicated for 81 2 min. Afterward, 500 mg of graphite microparticles and 125 mg of CNPs were added and the obtained mash 82 was sonicated for another minute. It was then immediately placed into the 5.7 mm inner diameter glass tube to 83 fill it to 3 mm in depth. After drying at room temperature for at least 48 h, it was polished with printing paper, 84 rinsed with water and dried. The geometric surface area of the exposed disc was 0.255 cm 2 . The electrodes were 85 immersed in 1 g L -1 solution of BOx in 0.1 M phosphate buffer pH 5.0 at +4 ºC for 2 h and rinsed with water.

86

For determination of dioxygen permeability of MTMOS-based silicate a hole (c.a. 100 µm diameter) in 0.

93

(Hilgenberg GmbH). Its apex was shaped according to literature procedure [40]. Optical micrographs were taken 94 using Nikon Eclipse LV150 microscope with LU Plan 10× objective (Nikon). All experiments were carried out 95 at ambient temperature and pressure. The in-house developed package MIRA was used for data analysis [41].

96

Cyclic voltammetry (CV) was performed with an Autolab PGSTAT 30 (Eco Chemie) electrochemical system 97 with dedicated software in a conventional three electrode cell with a Pt wire (0. micrograph is well focused on both materials (Fig. 1). This is also observed from the opposite side of the sample 116 (not shown). Therefore, one may conclude that the entire hole is filled with sol-gel processed silicate. Only two 117 polystyrene hills extend out of the focal plane. This is probably caused by a distorted laser beam focused using a 118 100 mm converging lens during manufacturing, because the incident beam is not perfectly parallel to the optical 119 axis of the lens accentuating coma and astigmatism. The height of these topographic defects is ca. 6 µm as 120 estimated using optical microscopy. These features are difficult to remove by polishing of the tiny polymer plate. 129 Subsequently, an image was recorded after the microelectrode potential had been switched to -0.4 V in order to 130 achieve diffusion-controlled dioxygen reduction at the Pt UME (Fig. 2B):

131

O 2 + 4H + + 4e -→ 2H 2 O (1)

132

A larger cathodic current is observed over the silicate filled hole (Fig. 2B) indicating that dioxygen concentration 135

In order to estimate dioxygen concentration above the silicate and the dioxygen flux, a cross section above the 136 centre of the silicate has been extracted from a SECM image (Fig. 3). The resulting profile has been fitted to the 137 following equations: 152

138 i T = 4nFD(O 2 ) r T [O 2 ] S ζ (2) 139 ( ) ( ) 2 S
Ω = 4D(O 2 ) r S [O 2 ] S = 2.66 × 10 -14 mol s -1 (4)
153

Although there is mass transfer resistance for dioxygen diffusion at the entrance of the pore from the gas phase 154 and at the exit from the pore to aqueous solution we supposed that is negligible as compared to that in the pore. [44]), the m for dioxygen transport in MTMOS-based silicate would be over two times higher than the 162 diffusion coefficient for dioxygen in water. Consequently, it can be concluded that the mass transport occurs 163 predominately over the silicate phase as gas diffusion in an interconnected pore system of the electrode material.

164

Next, the CCGE electrode was examined by SECM in K 4 [Fe(CN) 6 ] aqueous solution and the feedback mode was 165 used to visualise the local electrochemical reaction rate and the electronic conductivity of the CCGE (Fig. 4A).

166

Although the electrode was polished the topography can still affect this image [45]. First the microelectrode was 167 kept at 0.5 V in order to oxidize [Fe(CN) 6 ] 4-. Areas of higher feedback current (darker in Fig. 4 A) correspond to 168 the locally higher electronic conductivity. After switching the microelectrode potential to -0.4 V, the diffusion-169 controlled dioxygen reduction current at the microelectrode is recorded above the same region. Now darker areas 170 correspond to higher reduction currents (Fig. 4B) and represent these fragments of the CCGE surface where 171 more dioxygen reaches the electrode|electrolyte interface. Comparison of the SECM images in Figs. 4A and4B, 172 reveals that areas of higher conductivity are characterised by lower dioxygen supply. Vice versa, lower 173 conductivity areas correspond to higher dioxygen ingress. Areas with higher currents in Fig. 4A are most likely 174 graphite microparticles providing conductivity but being impermeable for dioxygen. The high currents fragments 175 shown in Fig. 4B are rich in silicates that provide diffusion paths for dioxygen, but have a lower conductivity.

176

The local analysis of transport pathways for electrons and gaseous reactants in composite electrodes provides a 177 new analytcial tool for optimising the composite structures of porous electrodes that may be important also for 178 other gas breathing electrodes.

179

The catalytic activity of the CCGE with adsorbed BOx was examined by cyclic voltammetry (Fig. 5) in 180 deaerated buffer. In this experiment dioxygen is supplied from the gas phase at the back side of the electrode.

181

The onset of the cathodic current is observed at 0.45 V at a slightly higher potential than for cobalt 182 tetraphenylporphyrin-modified CCGE (ca. 0.4 V vs. Ag|AgCl at pH 0) [5]. The cathodic current is proportional 183 to the dioxygen partial pressure at the back side of the electrode. (Fig. 5, lines 2 and 3). This dependence 184 observed at constant total pressure indicates that transport of the dioxygen from the gas phase affects the current 185 density of dioxygen bioelectroreduction. Interestingly, almost no hysteresis is observed in the voltammograms. 186 This is different than observed for analogous electrode operating in dioxygen-saturated solution [9]. This means 187 that the mass transport through the CCGE can maintain a steady state mass transport. (see ref. [12]). We had 188 already observed earlier that H 2 O 2 as a possible side product can be formed at CCGE, but only if the potential is 189 below 0.05 V vs. Ag|AgCl at pH 5 and dioxygen reduction occurs also at the carbon particles [9]. At higher 190 potentials at which all the oxygen reduction is catalyzed by BOx, no H 2 O 2 can be detected by SECM [9].

191

Finally, the CCGE was tested as cathode in a zinc-dioxygen primary cell [21][22][23]. A non-protected zinc wire 192 could be used as anode. Typical oxygen corrosion of zinc was avoided by using an essentially oxygen-free

203

The dioxygen mass transport rate through the porous carbon ceramic electrode consisiting of MTMOS-based 204 silicate is over two times higher than dioxygen diffusion in aqueous solution. However, the catalytic activity 205 does not differ much from the analogous CCE operated in dioxygen-containing aqueous solution [9]. The 206 construction of the electrode was not optimised and its efficiency as biocathode can be further improved by

  5 mm diameter) and a 98 Ag|AgCl|KCl sat. as auxiliary and reference electrode respectively. 99 Current density-voltage characteristics of the Zn-O 2 cell were determined by chronopotentiometry. The studied 100 cell acts as an active element of the electric circuit and the galvanostat is a passive element which allows the 101 current to flow with a certain setpoint value. The cell voltage for each current setpoint was recorded 100 s after 102 current setting. It was observed that after this period the voltage was stabilized. Then the galvanostat acts as 103 resistor with voltmeter and ammeter and results are comparable to those obtained by variable resistor method.104The advantage of chronopotentiometry is that it is fully automated.105All experiments were performed in deaerated electrolytes and argon was passed above the cells. nanoparticulate electrodes[12,13] indicates that the transport of dioxygen remains the rate 109 determining step of the mediatorless bioelectrocatalysis. Therefore permeation of dioxygen trough porous 110 MTMOS-based silicate matrix, being the gas permeable CCGE component is a crucial factor. In order to 111 visualize and estimate dioxygen permeability of this silicate material a hole (ca. 150 µm in diameter) in 0.8 mm 112 thick polystyrene plate filled with this material was studied by SECM. In order to provide the most shallow 113 depth of field the topography of the sample was characterized by optical microscopy with totally opened 114 aperture. Both silicate and polystyrene are at the same distance from objective of microscope, because the 115

  133 is higher comparing to other parts of the sample surface. This indicates the ingress of dioxygen from the 134 backside of the gel-closed hole into the deaerated electrolyte.

  developed for diffusion at an isolated disk-shaped pore[42,43], where n = 4 is the number 141 of transferred electrons from the microelectrode per O 2 molecule, D(O 2 ) = 2.0 × 10 -5 cm 2 s -1 [44] is the diffusion 142 coefficient of dioxygen, r T = 5 µm is the microelectrode radius, r S is the radius of the silicate hole, [O 2 ] S is the 143 additional concentration of dioxygen at the silicate surface, and ζ is a dimensionless factor describing the 144 decrease of dioxygen concentration as a function of the lateral distance ∆x = xx 0 and the vertical distance d 145 from the centre of the silicate spot. Non-linear curve fitting (Fig. 3) yields the following adjustable parameters: 146 4FDr T c S = 0.161 nA, x 0 = 275 µm, d = 10.1 µm, r S = 79.9 µm, i offset = 2.08 nA.147The fitted value of d agrees well with the distance the microelectrode was retracted from the surface before 148 recording the image. The value of r S is consistent with the 75 µm diameter estimated by optical microscopy. The 149 offset current is a consequence of the presence of dioxygen traces in the electrolyte. The additional dioxygen 150 surface concentration [O 2 ] S can be estimated as 0.0417 mM, i.e. 15.4 % of dioxygen concentration under air and 151 atmospheric pressure. From this value, the flux Ω of dioxygen from the silicate can be calculated[43]:

155

  Assuming that O 2 partition coefficient between silicate and aqueous solution is equal unity and uniform flux 156 through the silicate the mass-transfer coefficient m for dioxygen in MTMOS-based silicate can be l is mass-transport length equal to the sample thickness and [O 2 ] S ' is dioxygen surface concentration at the 160 back side of the sample. If one assume [O 2 ] S ' = 2.7 × 10 -5 mol cm -3 (corresponding to an air-saturated aqueous 161

  193 electrolyte. Current density-voltage and power density-voltage characteristics are presented in Fig 6A and 6B 194 respectively. The highest open-circuit voltage (1.095 V) was recorded when pure dioxygen was supplied to the 195 back side of the biocathode. The highest power density (41.3 µW cm -2 ) was recorded under the same conditions 196 at a cell voltage of 0.585 V. Clearly, the cell efficiency strongly depends on the external dioxygen partial shown that a carbon ceramic electrode composed of hydrophobic sol-gel processed silicate, carbon 200 microparticles and carbon nanoparticles exhibits bioelectrocatalysis of dioxygen reduction after adsorption of 201 bilirubin oxidase. When its back side is opened to a dioxygen-containing gas phase it can be operated as gas 202 electrode. The bioelectrocatalytic current is proportional to the partial pressure of dioxygen present in gas phase.

Scheme 1 :Fig. 1 :Fig. 2 :Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 :

 1123456 Scheme 1: A scheme of carbon ceramic gas electrode for bioelectrocatalytic reduction of dioxygen supplied

  Initially SECM imaging were performed in aqueous K 4 [Fe(CN) 6 ] solution with the microelectrode polarized at 124 +0.5 V to achieve a diffusion-controlled oxidation of [Fe(CN) 6 ] 4-(Fig.2A). The microelectrode current does not 125 depend on the microelectrode position except for the two small hills of polystyrene mentioned above. Over these 126 spots the current decrease due more efficient blocking of the diffusion of the [Fe(CN) 6 ] 4-to the microelectrode 127 above the impermeable elevated structures. The lack of any current difference between the position of the hole 128 (between the hills) and the surrounding PS plate indicates that the hole is completely filled with silicate.

	F o r
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121 Therefore, we decided to use them for SECM experiments bearing in mind that they can affect the obtained 122 currents.

We thank Dr. Yuriy Stepanenko and Bartlomiej Bialkowski (Institute of Physical Chemistry, Polish Academy of

Acknowledgements 215