Pedro Marinho
email: marinho@i3s.unice.fr

R De Oliveira

Vicente Zarzoso
email: zarzoso@i3s.unice.fr

Source Analysis and Selection Using Block Term Decomposition in Atrial Fibrillation

Keywords: Atrial Source Selection, Block Term Decomposition, Atrial Fibrillation, Blind Source Separation

Atrial fibrillation (AF) is the most common sustained cardiac arrhythmia in clinical practice, and is becoming a major public health concern. To better understand the mechanisms of this arrhythmia an accurate analysis of the atrial activity (AA) signal in electrocardiogram (ECG) recordings is necessary. The block term decomposition (BTD), a tensor factorization technique, has been recently proposed as a tool to extract the AA in ECG signals using a blind source separation (BSS) approach. This paper makes a deep analysis of the sources estimated by BTD, showing that the classical method to select the atrial source among the other sources may not work in some cases, even for the matrix-based methods. In this context, we propose two new automated methods to select the atrial source by considering another novel parameter. Experimental results on ten patients show the validity of the proposed methods.

Introduction

Atrial fibrillation (AF) is the most common sustained cardiac arrhythmia in clinical practice, responsible for up to 25% of strokes and 1/3 of the hospitalizations due to cardiac related disturbances [START_REF] January | AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: A report of the American College of Cardiology/American Heart Association Task Force on practice guidelines and the Heart Rhythm Society[END_REF]. This arrhythmia is becoming a major public health concern, since about 160 000 new AF cases are discovered every year only in USA, with similar numbers in European countries. This makes AF an increasingly prevalent disease that could become a new epidemic over the years [START_REF] Mainardi | Understanding Atrial Fibrilation: The Signal Processing Contribution[END_REF]. The mechanisms of this supraventricular arrhythmia are not completely understood, making AF a challenging cardiac condition, considered as the last great frontier of cardiac electrophysiology. During AF, electrical impulses typically generated around the pulmonary veins propagate in a chaotic and irregular way across the atria, replacing the P wave, that corresponds to a normal atrial activation, by low-amplitude fibrillatory waves, or f-waves. The f-waves are present through all the electrocardiogram (ECG) recording, but masked by the QRS complex of ventricular activity (VA) in each heartbeat.

To better understand the mechanisms of AF, it is necessary an accurate analysis of the atrial activity (AA), specifically, the f waves. A noninvasive analysis can be made by extracting the AA from the cardiac signals recorded by the standard 12-lead ECG using matrix decompositions techniques for blind source separation (BSS), such as principal component analysis (PCA) and independent component analysis (ICA) [START_REF] Rieta | Atrial activity extraction for atrial fibrillation analysis using blind source separation[END_REF][START_REF] Castells | Spatiotemporal blind source separation approach to atrial activity estimation in atrial tachyarrhythmias[END_REF][START_REF] Zarzoso | Extraction of ECG characteristics using source separation techniques: exploiting statistical independence and beyond[END_REF]. This matrix decomposition approach has proven to be useful for AA extraction. However, it has some limitations, since constraints need to be imposed to guarantee the uniqueness of such decompositions, e.g., orthogonality or statistical independence between the sources. Although mathematically coherent, such constraints may lack physiological grounds.

In order to overcome these limitations, a tensor approach has recently been proposed to analyze AF signals [START_REF] Ribeiro | Atrial signal extraction in atrial fibrillation electrocardiograms using a tensor decomposition approach[END_REF][START_REF] Ribeiro | A tensor decomposition approach to noninvasive atrial activity extraction in atrial fibrillation ECG[END_REF][START_REF] Ribeiro | Enhanced block term decomposition for atrial activity extraction in atrial fibrillation ECG[END_REF][START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF]. As compared to matrix techniques, tensor decompositions present some remarkable features such as essential uniqueness with practically minimal or no constraints. The block term decomposition (BTD) proposed in [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF] suits the characteristics of AA in an AF signal, since atrial signals can be approximated by all-pole models and mapped onto Hankel matrices with rank equal to the number of poles [START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF]. These Hankel matrices that contain the ECG data are stacked in the third dimension of a 3 rd -order tensor, and then processed by BTD. Previous experimental results in synthetic and real ECG data showed the potential superiority of BTD as compared to matrix decompositions for short ECG recordings [START_REF] Ribeiro | Atrial signal extraction in atrial fibrillation electrocardiograms using a tensor decomposition approach[END_REF][START_REF] Ribeiro | A tensor decomposition approach to noninvasive atrial activity extraction in atrial fibrillation ECG[END_REF][START_REF] Ribeiro | Enhanced block term decomposition for atrial activity extraction in atrial fibrillation ECG[END_REF].

The success of the BSS approach to AA extraction depends on the accurate identification of the atrial signal among the estimated sources. The classical method for atrial source selection consists in selecting the source with the highest spectral concentration (SC) among the sources whose dominant frequency (DF) lies between 3 and 9 Hz [START_REF] Rieta | Atrial activity extraction for atrial fibrillation analysis using blind source separation[END_REF][START_REF] Castells | Spatiotemporal blind source separation approach to atrial activity estimation in atrial tachyarrhythmias[END_REF]. The present work makes a deep analysis in the sources estimated by BTD, showing that the classical method may not work in some cases, even when the matrix-based approach is used. Taking this into account, a new parameter to improve the performance of the classical method is proposed. This parameter consists in analyzing the power of the source contribution to the lead V1, a lead that significantly reflects AA. Also, a new automated method for atrial source selection is proposed, using the proposed parameter and another one based on the kurtosis of the signal in the frequency domain. Experimental results using ten patients with persistent AF evaluate the proposed methods, showing their better performance in selecting the atrial source among the sources estimated by BTD and two matrix-based methods previously proposed in literature for AA extraction: RobustICA-f [START_REF] Zarzoso | Robust independent component analysis by iterative maximization of the kurtosis contrast with algebraic optimal step size[END_REF] and PCA [START_REF] Jolliffe | Principal Component Analysis[END_REF]. It is also shown that BTD can provide a better estimation of the AA signal, outperforming the matrix-based techniques in most cases.

The rest of this paper is organized as follows. Section 2 introduces the notation used in the work. Section 3 recalls the BTD as a tensor approach to solve BSS problems, while Section 4 discusses the estimated sources and the atrial source selection methods. Section 5 presents the experimental results and, fi-nally, Section 6 formulates the conclusion of this work, as well as the prospects for future works.

Notations

Scalars, vectors, matrices and tensors are represented by lower-case (a, b, c, ...), boldface lower-case (a, b, c, ...), boldface capital (A, B, C, ...) and calligraphic (A, B, C, ...) letters, respectively.

The transpose is represented by (•) T , symbol • represents the l 2 -norm and • represents the outer product. The operator diag(•) builds a diagonal matrix by placing its arguments along the diagonal. Given a 3 rd -order tensor A ∈ C I1×I2×I3 , with scalars a i1,i2,i3 , its frontal slices are represented by A ..i3 ∈ C I1×I2 . Given a matrix A ∈ C I1×I2 , with scalars a i1,i2 , its i th 1 row and the i th 2 column are represented by a i1. and a .i2 , respectively.

Block Term Decomposition

The BTD of an arbitrary 3 rd -order tensor T ∈ R I1×I2×I3 is written as

T = R r=1 E r • c r , (1)
with c r ∈ R I3 . Matrix E r ∈ R I1×I2 has rank L r and admits a decomposition E r = A r B T r , where A r ∈ R I1×Lr and B r ∈ R I2×Lr have rank L r .
We may then rewrite (1) as

T = R r=1 A r B T r • c r . (2)
One can see that the BTD is a decomposition of T in multilinear rank-(L r ,L r ,1) terms. If the matrix factors

A = A 1 A 2 . . . A R ∈ R I1× R r=1 Lr and B = B 1 B 2 . . . B R ∈ R I2× R r=1 Lr are full-column rank, which requires that R r=1 L r ≤ I 1 , I 2 , and C = c 1 c 2 . . . c R ∈ R I3×R
does not contain proportional columns, then the BTD is essentially unique [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF]Theorem 2.2]. Milder uniqueness conditions are presented in [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF].

The ECG data matrix, with K leads and N samples, can be modeled as

Y = MS ∈ R K×N , (3)
where M ∈ R K×R is the mixing matrix, modelling the propagation of the cardiac electrical sources from the heart to the body surface, S ∈ R R×N is the source matrix that contains the atrial, ventricular and noise sources, and R is the number of sources [START_REF] Zarzoso | Extraction of ECG characteristics using source separation techniques: exploiting statistical independence and beyond[END_REF]. The AA extraction in an AF ECG recording can be seen as a BSS problem, since the only data observed is matrix Y, and we aim to estimate M and S from it. In [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF], the BTD is proposed as a solution of a BSS problem like (3), but does not deal with the AA extraction specifically.

The idea to obtain a tensor from Y is to map its k th row onto a Hankel matrix

H (k) Y ∈ R I×J , where I = J = N +1 2 if N is odd or I = N 2 and J = N 2 + 1 if N is even, with [H (k) Y ] i,j y k,i+j-1 , (4)
where i = 1, ..., I, j = 1, ..., J, and k = 1, ..., K. Next, the tensor is built by stacking each Hankel matrix along the third dimension (as frontal slices) of a 3 rd -order tensor Y ∈ R I×J×K , that is

Y ..k = H (k) Y . (5)
The k th matrix slice of the tensor Y can be represented as

Y ..k = R r=1 m k,r H (r) S , (6)
where

H (r)
S ∈ R I×J is a Hankel matrix built from the r th row of S. We can see that for each r, the outer product between matrix H (r) S and the r th column of M, i.e., m .r , is being performed. This way, the tensor Y can be written as

Y = R r=1 H (r) S • m .r . (7)
Comparing Equation ( 1) with ( 7), we can conclude that the tensor ECG data follows a BTD tensor model.

During AF, the AA presents certain harmonicity. Hence, atrial sources can plausibly be represented by the exponential (or all-pole) model as

s r,n = Lr =1 λ ,r z n-1 ,r , (8)
where n = 1, ..., N , r = 1, .., R, L r is the number of exponential terms, z ,r is the th pole of the r th source, and λ ,r is the scaling coefficient [START_REF] Ribeiro | Atrial signal extraction in atrial fibrillation electrocardiograms using a tensor decomposition approach[END_REF][START_REF] Ribeiro | A tensor decomposition approach to noninvasive atrial activity extraction in atrial fibrillation ECG[END_REF][START_REF] Ribeiro | Enhanced block term decomposition for atrial activity extraction in atrial fibrillation ECG[END_REF][START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF]. This way, their associated Hankel matrix accepts the Vandermonde decomposition as in [START_REF] Boley | Vandermonde factorization of a Hankel matrix[END_REF].

4 Atrial Source Selection

Classical Method

To select the AA signal or the source with the most significant AA activity, the classical method considers two parameters. The first one is the DF, that is, the position of the peak frequency in the power spectral density, since the AA during AF typically has a peak between 3 and 9 Hz. The second parameter, called SC, is the relative amount of energy around the DF, and it is calculated as:

SC = 1.17fp 0.82fp P AA (f i ) Fs/2 0 P AA (f i ) , (9)
where f p is the value of DF, F s is the sampling frequency and P AA is the power spectrum of the AA signal, estimated as in [START_REF] Castells | Spatiotemporal blind source separation approach to atrial activity estimation in atrial tachyarrhythmias[END_REF]. In this work, the SC is calculated over the first harmonic (fundamental frequency) only.

The classical method of atrial source selection makes the assumption that the atrial source is concentrated in a single source only. This method consists of the following steps:

1. Select all the estimated sources with DF between 3 and 9 Hz. We refer to sources fulfilling this condition as potential atrial sources. 2. Select the potential atrial source with the highest SC.

Proposed Method 1

In the literature, the classical automated method described above has been used to detect the atrial source among the other estimated sources. However, in some cases, this method may not precisely select the atrial source, as will be illustrated later in this work. In Figures 3 and4, for example, the atrial source does not correspond to the potential source with the highest SC, despite the fact that they have close values of SC at very close DF positions.

Aiming at a better estimation of the AA signal, this paper proposes two new parameters. The first new parameter is the power contribution to the recording, which is given by

P (r) = 1 N ||m (V 1) r s r. || 2 , ( 10
)
in mV 2 , where m

(V 1) r
is the contribution of the r th source to lead V1 (given by the corresponding element of the estimated mixing matrix) and s r. is the r th source, corresponding to the r th row of matrix S in Equation (3). Using the power contribution to the recording as a new parameter the classical method becomes:

1. Select all the estimated sources with DF between 3 and 9 Hz (potential atrial sources). 2. Select all the potential atrial sources with power contribution higher than 10 -4 mV 2 . We refer to this subset of sources as likely atrial sources. 3. Select the likely atrial source with the highest SC.

Selecting the sources with power contribution higher than 10 -4 mV 2 is needed in order to eliminate all sources that may present AA-like signature but are actually too weak to represent AA components. This threshold is chosen based on initial experiments that showed that sources with power contribution lower than 10 -4 mV 2 do not present any significant contribution to the original signal. The power contribution is calculated in lead V1 due to the fact that this lead typically reflects AA best in AF ECGs, as its exploring electrode lies close to the right atrium.

Proposed Method 2

In order to better select the source with the most significant AA content among the other estimated sources, a new automated method is now proposed. The first two steps of this method are the same as those of the proposed method introduced in the previous subsection. The third and last step of this new method is to compute the kurtosis, denoted K, of the signal in the frequency domain acquired by a 4096-point FFT (the second new parameter). As in [START_REF] Zarzoso | Robust independent component analysis by iterative maximization of the kurtosis contrast with algebraic optimal step size[END_REF], we use the general expression of kurtosis valid for non-circular complex data. The likely atrial source with the highest kurtosis is related as the atrial source.

In the experiments below, it will be shown that selecting the source with the highest kurtosis provides a better performance than selecting the source with the highest SC. A possible explanation is that kurtosis is computed from the whole signal, while SC is only computed around the DF. Recall that AA in AF is typically a harmonic signal, characterized by a sparse frequency spectrum with few values significantly different from zero. Kurtosis is a measure of peakedness and sparsity of a distribution and, when computed in the frequency domain, it naturally provides a quantitative measure of harmonicity of the signal. Also, kurtosis is parameter free, whereas SC depends on the DF and the definition of a suitable interval for interpretation.

Experimental Results

Real AF ECG Data and Preprocessing

The recordings used in our experiments belong to a database provided by the Cardiology Department of the Princess Grace Hospital Center, Monaco. These recordings were acquired at a 977 Hz sampling rate and preprocessed by a zero-phase forward-backward type-II Chebyshev bandpass filter with cutoff frequencies of 0.5 and 40 Hz, to suppress high-frequency noise and baseline wandering. To analyze the potential atrial sources, we consider a randomly selected heartbeat (QRS-T complex + TQ segment) of a standard 12-lead ECG recording from a persistent AF patient. A single-beat segment of this patient is shown in Figure 1, where we can see the TQ interval just after the QRS-T complex in lead II. The beat from this patient used to analyze the potential atrial sources is chosen randomly and has 1300 samples.

To asses the atrial selection methods, a population of 10 patients with persistent AF is used in the same way previously described. Similarly, one beat from each of the ten patients is chosen randomly to evaluate atrial source selection performance. The lengths of the chosen beats is between 1000 and 1400 samples (1.02 and 1.43 seconds, respectively). Due to lack of space, the potential source analysis of all ten patients is not reported in this paper. So only the first patient of the observed population was chosen for source analysis.

BTD Setup

The BTD is implemented using the non-linear least squares (NLS) method available in Tensorlab MATLAB toolbox [START_REF] Vervliet | [END_REF] for r = 1, 2, ..., 12. This choice is made based on the work [START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF], which showed that these values provided good results for the heartbeat with the largest TQ segment of one of the patients in the present observed population. The tolerance threshold for convergence is set to 10 -9 and the maximum number of iterations is set to 1000. BTD is known to be dependent on a suitable initialization of its factors. The experiments reported in this section evaluate the influence of BTD factors initialization on source estimation performance and atrial source selection. Ten Monte Carlo runs, with normalized Gaussian random initialization for the matrix and vector factors at each run, are used to analyze the potential atrial sources found by BTD and compare them with the ones found by the matrix-based methods PCA and RobustICA-f. All the beats are downsampled by a factor of two, since the 3 rd -order tensor built from the original 12-lead ECG beat poses some difficulties to Tensorlab.

Potential Atrial Source Analysis

For the observed patient used to analyze the potential atrial sources, PCA found 6 potential sources, RobustICA-f found 5 potential sources and BTD found a mean of 7.2 potential sources. In 7 out of the 10 independent runs, the BTD found more potential sources than the matrix-based methods, reflecting the ability of the tensor technique to perform undertermined source separation [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF]. Finding several potential atrial sources is interesting, since it increases the possibility of finding some features that, although weakly contributing to the AA, may provide useful physiological and clinical information about the arrhythmia. In this work, however, we assume as in the previous literature of this topic that all AA can be represented by a single source, and we leave the multiple source hypothesis for further works. Due to the lack of space and for the sake of clarity, only two potential atrial sources for PCA, RobustICA-f and BTD are shown in Figures 234. The other sources were disregarded for presenting a very weak power contribution.

Looking at Figure 2, we can see that the atrial source estimated by PCA (located in the second row) has SC equal to 62.5%, while looking at Figure 3, the estimated atrial source by RobustICA-f (located in the second row) has SC equal to 68.3%. For BTD, 8 out of the 10 independent runs estimated an atrial source with higher SC than PCA and 6 with higher SC than both matrix-based methods, giving an average SC over the 10 runs equal to 67.8%. Figure 4 shows the results for a particular initialization of BTD, where the estimated atrial source (located in the second row) has SC equal to 76.5%. The DF position of both PCA and RobustICA-f are located at 5.96 Hz, while in BTD it lies between 5.72 and 5.96 Hz. For comparison, the DF position obtained from an electrogram simultaneously acquired by a catheter located in the left atrial appendage of the same patient, is equal to 4.77 Hz.

Atrial Source Selection

As ground truth, the sources were visually analyzed in time and frequency domain with guidance of the parameters previously described. The source with the strongest representation of AA content was taken as the atrial source.

The classical method and the two proposed methods of atrial source selection were assessed in 10 segments of 10 different patients, as previously explained. From a total of 120 runs for the 10 patients (100 for BTD, 10 for PCA and 10 for RobustICA-f) the classical method succeeded only in 45.8% of runs. Applying the first proposed method, the index of success increases to 75%, while the second proposed technique succeeds in 83.7% of the trials. It should be mentioned that in 35.8% of the trials, both the classical and the second proposed method were able to select the source with most AA content. Also, in 12.5% of trials none of the methods were able to select the AA signal. means that the existing methods are suboptimal regarding the AA source selection. However, from the reported experiments, it is believed that a balanced combination between power contribution and kurtosis may lead to an optimal or at least a better method.

Conclusions

The present work has analyzed the potential atrial sources estimated by BTD, showing its satisfactory performance for most initializations in the tested database. We have shown that the classical method of atrial source selection may not work in some cases, and we have proposed two new automated methods that better select the atrial source among the other potential sources. These methods have been validated in experimental results not only for BTD but also for the matrix-based methods PCA and RobustICA-f in a population of 10 patients with persistent AF. In future works, we aim to assess the proposed methods in a larger database and along consecutive time segments of each patient to analyze intra-patient (temporal) stability.

Fig. 1 :

 1 Fig. 1: A single-beat segment of an AF ECG recording of one patient in lead II. The vertical line marks the location of the T-wave offset.

Fig. 2 :

 2 Fig. 2: Potential atrial sources contribution to lead V1 estimated by PCA. Left: time domain (in mV). Right: frequency domain (in mV/ √ Hz).

Fig. 3 :Fig. 4 :

 34 Fig. 3: Potential atrial sources contribution to lead V1 estimated by RobustICAf. Left: time domain (in mV). Right: frequency domain (in mV/ √ Hz).

Doctoral School of the Université Côte d'Azur. Vicente Zarzoso is a member of the


Pedro Marinho

R De Oliveira

Vicente Zarzoso

Block Term Decomposition Analysis in Long Segments of Atrial Fibrillation ECGs

Keywords: Atrial Fibrillation, Block Term Decomposition, Electrocardiogram, Atrial Activity Extraction, Tensor Decomposition

Responsible for 25% of strokes, atrial fibrillation (AF) is the most common sustained cardiac arrhythmia in clinical practice. A precise analysis of the atrial activity (AA) signal in electrocardiogram (ECG) recordings is necessary to better understand this challenging cardiac condition. Recently, the block term decomposition (BTD) has been proposed as a novel tool to extract AA in AF ECG signals noninvasively. However, this tensor factorization technique was performed only in short segments. In this paper, the BTD is assessed in long segments of an AF ECG, varying the observation window size. Experimental results show the performance of BTD in long segments of an AF ECG recording and the analysis of the observation window size. This promising tensor technique is compared to two matrix-based methods.

I. INTRODUCTION

Cardiac arrhythmia is a disease characterized by an irregular heartbeat. Especially affecting the eldery, atrial fibrillation (AF) is the most common sustained cardiac arrhythmia encountered in clinical practice, considered as the last great frontier of cardiac electrophysiology. Persistent AF represents a highly complex case of this arrhythmia, where extensive atrial remodelling has taken place due to sustained AF, significantly affecting atrial activity (AA) and AF perpetuation itself. The importance given to this challenging cardiac condition has increased in the past few years and is expected to increase even more, since it is becoming a major public health and economical concern, and its mechanisms are not completely understood. About 160 000 new patients are diagnosed with AF every year only in USA and similar numbers are shown in European countries. By 2050, it is estimated that AF will become a new epidemic [START_REF] Mainardi | Understanding Atrial Fibrilation: The Signal Processing Contribution[END_REF]. In addition, a patient with AF spends anually approximately $8 700 more in healthcare than a patient without AF. The treatment of this decease is estimated to add $26 billion per year to the USA healthcare bill [START_REF] January | AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: A report of the American College of Cardiology/American Heart Association Task Force on practice guidelines and the Heart Rhythm Society[END_REF]. During a normal sinus rhythm in the electrocardiogram (ECG), the P wave corresponds to a normal atrial activation. During AF, the P wave is replaced by f waves (fibrillatory waves), which are present throughout the whole ECG recording. However, they are masked by the QRST complex that corresponds to the ventricular activity (VA) in each heartbeat Signal processing techniques are useful and important tools in the analysis of cardiac signals. They are necessary to noninvasive extract the AA from the standard 12-lead ECG Pedro Marinho R. de Oliveira and Vicente Zarzoso are with the Université Côte d'Azur, I3S Laboratory, CNRS, France. (e-mail: {marinho, zarzoso}@i3s.unice.fr).

for an accurate analysis of the f waves, in order to better understand the mechanisms behind this disease. Matrix-based techniques to solve blind source separation (BSS) problems, such as principal component analysis (PCA) and independent component analysis (ICA) [START_REF] Rieta | Atrial activity extraction for atrial fibrillation analysis using blind source separation[END_REF]- [START_REF] Zarzoso | Extraction of ECG characteristics using source separation techniques: exploiting statistical independence and beyond[END_REF] have proved to be useful in AA extraction. However, matrix decompositions are known to have some limitations since constraints need to be imposed to guarantee uniqueness, e.g., orthogonality or statistical independence between the sources. Such constraints are mathematically coherent, but they may lack physiological grounds, making difficult the results interpretation.

Aiming to overcome the limitations imposed by matrix decompositions, a tensor factorization technique has been recently proposed to extract the AA signal from AF ECGs [START_REF] Ribeiro | Atrial signal extraction in atrial fibrillation electrocardiograms using a tensor decomposition approach[END_REF]- [START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF]. If compared to matrix-based techniques, tensor decompositions present some remarkable features like, for example, their essential uniqueness with minimal or no constraints. Another example is the fact that the rank of the tensor can exceed its dimensions, whereas in matrices the rank is limited by its lowest dimension. The block term decomposition (BTD) proposed in [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF] suits the characteristics of AA in an AF signal, since atrial signals can be approximated by all-pole models and mapped onto Hankel matrices with rank equal to the number of poles [START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF]. The Hankel matrices containing the ECG data are stacked along the third dimension of a third-order tensor, and then processed by BTD. Previous experimental results in synthetic and real AF ECG data showed that BTD can outperform the matrix-based techniques for AA extraction in short segments of AF ECG recordings [START_REF] Ribeiro | Atrial signal extraction in atrial fibrillation electrocardiograms using a tensor decomposition approach[END_REF]- [START_REF] Ribeiro | Enhanced block term decomposition for atrial activity extraction in atrial fibrillation ECG[END_REF].

The performance of BTD depends considerably on its initialization and model parameters, especially the multilinear rank (the rank of the matrix factors) and the number of block terms of the tensor. In [START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF] an estimation method of the multilinear rank of BTD in an AF ECG tensor is proposed. This estimation is based on the application of three popular methods for the estimation of auto-regressive (AR) model order in the TQ segment of a heartbeat, where only AA is present. In addition, a sucessful AA extraction depends on the precise selection of the AA source among the others estimated sources. In [START_REF] De Oliveira | Source analysis and selection using block term decomposition in atrial fibrillation[END_REF] the BTD showed that the classical method of atrial source selection [START_REF] Rieta | Atrial activity extraction for atrial fibrillation analysis using blind source separation[END_REF], [START_REF] Castells | Spatiotemporal blind source separation approach to atrial activity estimation in atrial tachyarrhythmias[END_REF] is not optimal, and then two suboptimal methods that provide better accuracy were proposed.

The BTD has proved to be useful extracting the AA signal from the AF ECG, being able to outperform the matrixbased techniques. However, this tensorial approach was only assessed in short segments of an AF ECG (1 to 5 seconds). Its performance in long segments remains an open question.

To answer this question and analyze the observation window size on the performance of this new promising technique, the present work assesses BTD for the first time in long segments of a patient with persistent AF and compares it to two popular matrix-based methods for AA extraction: RobustICAf [START_REF] Zarzoso | Robust independent component analysis by iterative maximization of the kurtosis contrast with algebraic optimal step size[END_REF] and PCA [START_REF] Jolliffe | Principal Component Analysis[END_REF]. Experimental results using Monte Carlo simulations evaluate the AA extraction performance of BTD on long segments of an AF ECG.

The rest of this paper is organized as follows. Section II introduces the notation used in the work, while Section III recalls the BTD as a tensor approach to solve BSS problems. Section IV discusses quantitative measures of AA content. Section V presents the database and the setup used in the experiments, whose results are reported in Section VI. Finally, Section VII gives an end to the paper with the conclusion of this work and the prospects of future works.

..i3 ∈ C I1×I2 . Given a matrix A ∈ C I1×I2
, with scalar entries a i1,i2 , its i th 1 row and the i th 2 column are represented by a i1. and a .i2 , respectively.

III. BLOCK TERM DECOMPOSITION

The BTD of an arbitrary third-order tensor T ∈ R I1×I2×I3 is written as

T = R r=1 E r • c r (1)
where c r ∈ R I3 and E r ∈ R I1×I2 has rank L r and admits a decomposition E r = A r B T r , where A r ∈ R I1×Lr and B r ∈ R I2×Lr have full column rank L r . Equation (1) may then be rewritten as

T = R r=1 A r B T r • c r . (2)
One can see that the BTD is a decomposition of T in multilinear rank-(L r ,L r ,1) terms. In [13, Theorem 2.2], it is shown that the BTD is essentially unique if the following conditions are satisfied:

1) The matrix factors

A = A 1 A 2 . . . A R ∈ R I1× R r=1 Lr and B = B 1 B 2 . . . B R ∈ R I2× R r=1 Lr are full-column rank. 2) Matrix C = c 1 c 2 . . . c R ∈ R I3×R does not
contain proportional columns. The first condition requires that R r=1 L r ≤ I 1 , I 2 . Milder uniqueness conditions are also presented in [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF].

The ECG data matrix with K leads and N samples can be modeled as

Y = MS ∈ R K×N (3)
where M ∈ R K×R is the mixing matrix, modeling the propagation of the cardiac electrical sources from the heart to the body surface, S ∈ R R×N is the source matrix that contains the atrial, ventricular and noise sources, and R is the number of sources [START_REF] Zarzoso | Extraction of ECG characteristics using source separation techniques: exploiting statistical independence and beyond[END_REF]. Since the goal is to estimate M and S from the matrix Y (the only observed data), it is clear that AA extraction in an AF ECG recording is a BSS problem. In [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF], the BTD is proposed as a solution of a BSS problem like [START_REF] Rieta | Atrial activity extraction for atrial fibrillation analysis using blind source separation[END_REF].

The idea to obtain a tensor from Y is to map its k th row onto a Hankel matrix

H (k) Y ∈ R I×J , where I = J = N +1 2 if N is odd or I = N 2 and J = N 2 + 1 if N is even, with [H (k) Y ] i,j y k,i+j-1 (4)
where i = 1, ..., I, j = 1, ..., J, and k = 1, ..., K. Next, the tensor is built by stacking each Hankel matrix along the third dimension (as frontal slices) of a third-order tensor Y ∈ R I×J×K , that is

Y ..k = H (k) Y . (5)
Tensor Y can be written in scalar form as

y i,j,k = R r=1 m k,r s r,i+j-1 . (6)
The k th matrix slice of this tensor can be represented as

Y ..k = R r=1 m k,r H (r) S (7)
where

H (r)
S ∈ R I×J is a Hankel matrix built from the r th row of S. For each r, the outer product between matrix H (r) S and the r th column of M, i.e., m .r , is being performed. This way, the tensor Y can be written as

Y = R r=1 H (r) S • m .r . (8)
Comparing Equations ( 1) and ( 8), it can be concluded that the ECG data tensor follows a BTD tensor model with the following correspondence

(T ) ⇐⇒ (Y) (9)
(E r , c r ) ⇐⇒ (H (r) S , m .r ) (10)
(I 1 , I 2 , I 3 , R) ⇐⇒ (I, J, K, R) . (11)
Due to the quasi-periodic nature of AF, atrial sources can be represented by the exponential (or all-pole) model as

s r,n = Lr =1 λ ,r z n-1 ,r (12)
where n = 1, ..., N , r = 1, .., R, L r is the number of exponential terms, z ,r is the th pole of the r th source, and λ ,r is the scaling coefficient. This way, their associated Hankel matrices accept the Vandermonde decomposition [START_REF] Boley | Vandermonde factorization of a Hankel matrix[END_REF]:

H (r) S = V r diag(λ 1,r , λ 2,r , ..., λ Lr,r ) VT r (13)
with

V r =      1 1 . . . 1 z 1,r z 2,r . . . z Lr,r . . . . . . . . . . . . z I-1 1,r z I-1 2,r . . . z I-1 Lr,r      ∈ R I×Lr (14)
and

Vr =      1 1 . . . 1 z 1,r z 2,r . . . z Lr,r . . . . . . . . . . . . z J-1 1,r z J-1 2,r . . . z J-1 Lr,r      ∈ R J×Lr . (15)
In the case of different poles, the Vandermonde matrix with L r ≤ I, J will have full-column rank L r , so if M does not have proportional columns, the BTD in (8) will be essentially unique. In the case of equal poles, milder conditions can assure the uniqueness of (8) [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF].

As previously described, BTD can outperform the matrix approaches regarding the AA extraction in short AF ECG recordings [START_REF] Ribeiro | Atrial signal extraction in atrial fibrillation electrocardiograms using a tensor decomposition approach[END_REF]- [START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF]. However, its performance in long AF ECG segments remains an open question, since, to the best of the authors' knowledge, the BTD has only been computed in short segments of AF ECGs.

IV. ATRIAL ACTIVITY CONTENT MEASUREMENT

Signal processing techniques used to solve BSS problems separate the original signal in several sources. In AF ECG, typically at least one of these sources contains the AA. However, it is unknown if the AA is concentrated only in a single source. In the present work, as in previous works, it is considered that the AA is concentrated in a single source, and the source with the most AA content is called the atrial source. In the frequency domain, the AA during AF has a peak between 3 and 9 Hz. The position of this peak is called dominant frequency (DF). It is defined as potential atrial sources any source with DF placed in such an interval. Three parameters are used to measure AA extraction quality. In this work, the atrial source is selected by visual inspection helped by the parameters presented in this section. The first one is the spectral concentration (SC), that is, the relative amount of energy around the DF. The SC is computed as in [START_REF] Castells | Spatiotemporal blind source separation approach to atrial activity estimation in atrial tachyarrhythmias[END_REF]:

SC = 1.17fp fi=0.82fp P AA (f i ) Fs/2 fi=0 P AA (f i ) (16)
where f p is the value of the DF, F s is the sampling frequency, f i is the discrete frequency and P AA is the power spectrum of the AA signal computed using Welch's method as in [START_REF] Castells | Spatiotemporal blind source separation approach to atrial activity estimation in atrial tachyarrhythmias[END_REF]. The second parameter used to better analyze the potential atrial sources is the kurtosis, denoted K, of the signal in the frequency domain acquired by a 4096-point FFT. As in [START_REF] Zarzoso | Robust independent component analysis by iterative maximization of the kurtosis contrast with algebraic optimal step size[END_REF], it is used the general expression of kurtosis valid for non-circular complex data, that is given by

K = E[|s r. | 4 ] -2E[|s r. | 2 ] 2 -|E[s 2 r. ]| 2 E[|s r. | 2 ] 2 . ( 17
)
Kurtosis is a measure of peakedness and sparsity of a distribution and, when computed in the frequency domain, it naturally provides a quantitative measure of harmonicity of the signal. A high kurtosis in the frequency domain means that the power spectral density is sparse, which is suggestive of a harmonic signal like AA in AF. Also, kurtosis is parameter free, whereas SC depends on the DF and the definition of a suitable interval for interpretation [START_REF] De Oliveira | Source analysis and selection using block term decomposition in atrial fibrillation[END_REF]. The third parameter, used to eliminate sources with irrelevant content, is the power contribution to lead V1 given by

P (r) = 1 N ||m (V 1) r s r. || 2 (18)
in mV 2 , where m

(V 1) r
is the contribution of the r th source to lead V1 (given by the corresponding element of the estimated mixing matrix) and s r. is the r th source, corresponding to the r th row of matrix S in Equation ( 3). The power contribution to lead V1 by an AA source is expected to be strong, since this lead is the one that typically reflects best AA in AF ECGs.

V. DATABASE AND EXPERIMENTAL SETUP

Experiments on real AF ECG data evaluate the performance of BTD regarding the AA extraction in long segments varying the observation window size.

A. Real AF ECG Data and Preprocessing

Experiments are performed in 4 segments of a standard 12-lead ECG recording from a single patient suffering from persistent AF, considering all the 12 leads. These recordings belong to a database provided by the Cardiology Department of Princess Grace Hospital Center, Monaco. The recordings are acquired at a 977 Hz sampling rate and are preprocessed by a zero-phase forward-backward type-II Chebyshev bandpass filter with cutoff frequencies of 0.5 and 40 Hz, in order to suppress high-frequency noise and baseline wandering. A 15second segment in lead II from the patient is shown on Figure 1. The segments have about 2.5, 5, 10 and 15 seconds and they all have the same starting sample. They are downsampled by a factor of 4, 8, 16 and 24, respectively, since the originally built third-order tensors pose some difficulties to Tensorlab MATLAB toolbox [START_REF] Vervliet | [END_REF].

B. BTD Setup

The BTD is implemented using the non-linear least squares (NLS) method available in Tensorlab choosing R = 12 and L r = 95, for all r = 1, 2, ..., R. This choice is made based on the work [START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF], that shows that such values provide satisfactory results for the heartbeat with the largest TQ segment of the patient considered in the present work. The tolerance threshold for BTD convergence is set to 10 -9 and the maximum number of iterations is set to 1000. Ten Monte Carlo runs, with Gaussian random initialization for the matrix and vector factors at each run, are used to analyze the performance of BTD in the AF patient regarding AA signal extraction. Monte Carlo runs are needed, since BTD has a strong dependence on its initialization. This tensorial technique is compared to the matrix-based methods PCA and RobustICA-f.

A. Long Segments Performance

Table I shows the values of SC in % for PCA and RobustICA-f. The mean of the ten independent runs and the maximum value is shown for BTD. It can be seen in Table I that the mean of the SC for BTD is very close to the matrix-based techniques, outperforming PCA in all the observed segments and RobustICA-f in 3 out of the 4 segments with different lengths. Also, BTD is superior for all observed segments in 5 to 8 out of the 10 independent runs, showing that it can easily have superior performance if the right model parameters (R, L r ) and initialization are chosen. It can be seen also that the maximum SC presents a high value (≥ 75%) for all the observed segments, and in all cases, superior to that of matrix techniques, meaning that, with the right initialization, BTD outperformed to the matrix-based techniques for the considered window lengths.

In the observed segments of different lengths, BTD finds more potential atrial sources than PCA and RobustICA-f. Discovering many potential atrial sources may be an interesting outcome, since it increases the possibility of finding some features that, although weakly contributing to the AA, may provide useful physiological and clinical information about the arrhythmia.

Figures 2 and3 illustrate the estimated atrial source by the three BSS techniques compared in this paper, in the time and frequency domain, respectively, for the best performance of BTD in the 15-second segment. These two figures show the satisfactory performance of BTD in estimating the AA in long segments of an AF ECG, as well as its superiority compared to the matrix-based methods, as quantified by the higher SC and kurtosis values.

B. Observation Window Size Analysis

The observation window size used in the experiments extends from 2.5 to 15 seconds, in order to analyze its influence on BTD performance. Figures 4 and5 show the variation of SC and kurtosis, respectively, over the 10 independent runs for all the window observation sizes analyzed in this work. This variation is illustrated by a box-and-whisker plot, where the red lines represent the median and the blue boxes the percentiles. Due to the lack of space, similar figures for PCA and RobustICA-f are not shown in this paper.

It can be seen that there exists a certain variation of SC and kurtosis over the runs, for each observation window size. This is expected, since BTD performance depends considerably on the initialization of its model factors. However, there is no clear trend in the computed parameters, which shows that the influence of the observation window size on the performance of BTD is not very significant. The only drawback observed when processing long segments of ECG using BTD is the fact that the original segment must be downsampled by high factors, since Tensorlab has some difficulties in processing a tensor with large dimensions. Downsampling by a high factor could cause some loss of information in the signal.

VII. CONCLUSION

AF is becoming a major public health and economical concern. Signal processing techniques to solve BSS problems arise as a key tool to better understand the mechanisms behind this challenging cardiac condition. Aiming to overcome the limitations of matrix-based methods, a tensorial technique has been recently proposed as a useful tool for atrial signal extraction in AF ECGs. The first contribution of the present work is the analysis of BTD in time segments with varying length of an AF ECG, showing that this tensor technique presents a satisfactory performance not only in short segments (2.5 s), but also in long segments (15 s). The second contribution of the present work is the analysis of the observation window size on the performance of BTD. The choice of the size of the observation window is shown to be not very significant regarding the quality of the atrial signal extracted. The only drawback observed is the need of downsampling the long segment by a high factor, which may cause some loss of information. Future works would aim at analyzing the temporal (or intra-patient) stability of BTD. Experiments should be performed in a large database of AF patients to validate these results.

  II. NOTATIONS Scalars, vectors, matrices and tensors are represented by lower-case (a, b, c, ...), boldface lower-case (a, b, c, ...), boldface capital (A, B, C, ...) and calligraphic (A, B, C, ...) letters, respectively. The matrix transpose is represented by (•) T , symbol || • || represents the l 2 -norm, | • | represents the absolute value and • represents the outer product. E[•] denotes the mathematical expectation and the operator diag(•) builds a diagonal matrix by placing its arguments along the diagonal. Given a thirdorder tensor A ∈ C I1×I2×I3 , with scalars a i1,i2,i3 , its frontal slices are represented by A

Fig. 1 :

 1 Fig. 1: A 15-second segment of an AF ECG recording from one patient. Only bipolar limb lead II is shown, for clarity, although the full 12 leads are processed.

2 Fig. 2 :Fig. 3 :

 223 Fig. 2: Estimated atrial sources contribution to lead V1 in the 15-second segment by BTD, RobustICA-f and PCA in the time domain (in mV). AA signal estimations are vertically shifted for clarity.

Fig. 4 :

 4 Fig. 4: SC (%) of the AA signals estimated by BTD from the observed AF ECG segments over ten independent runs.

Fig. 5 :

 5 Fig. 5: Kurtosis of the AA signals estimated by BTD from the observed AF ECG segments over ten independent runs.

TABLE I :

 I Values of SC (%) for PCA and RobustICA-f. For BTD, the maximum (BTD max ) and the mean (BTD mean ) values of SC (%) of ten independent runs is shown.

		BTDmax	BTDmean	PCA	RobustICA-f
	2.5 s	94.04	74.03	55.29	69.08
	5 s	77.88	65.56	49.03	66.13
	10 s	90.93	73.49	56.40	72.22
	15 s	93.01	77.87	54.36	71.80
			VI. RESULTS

ACKNOWLEDGEMENTS

Pedro Marinho is funded by a PhD scholarship from the IT Doctoral School of the Université Côte d'Azur. Vicente Zarzoso is a member of the Institut Universitaire de France.


Pedro Marinho
email: marinho@i3s.unice.fr

R De Oliveira

Vicente Zarzoso
email: zarzoso@i3s.unice.fr

Temporal Stability of Block Term Decomposition in Noninvasive Atrial Fibrillation Analysis

Considered as the last great frontier of cardiac electrophysiology, atrial fibrillation (AF) is the most common sustained cardiac arrhythmia in clinical practice. The mechanisms behind this disease are not completely understood, and a precise analysis of the atrial activity (AA) signal is necessary to a better understanding. The block term decomposition (BTD) has been recently proposed as a useful tool for noninvasive AA extraction in electrocardiogram (ECG) signals. However, this tensor factorization technique was assessed only in short fixed segments of an AF ECG. To bridge this gap, the present work evaluates for the first time the temporal (intra-patient) stability of BTD by considering consecutive segments through a whole AF ECG recording. Experimental results show the consistency of BTD as an AA extraction tool, being capable of outperforming two well-known matrix-based methods and presenting a stable performance over time.

I. INTRODUCTION

Atrial fibrillation (AF) is the most common sustained cardiac arrhythmia encountered in clinical practice and is considered a major health and economical concern, since about 160 000 new patients are diagnosed with AF every year in USA only, and similar numbers are shown in some European countries [START_REF] Mainardi | Understanding Atrial Fibrilation: The Signal Processing Contribution[END_REF]. Also, a patient with AF costs anually approximately $8 700 more in healthcare than a patient without AF. The treatment of this disease is estimated to add $26 billion per year to the USA healthcare bill [START_REF] January | AHA/ACC/HRS guideline for the management of patients with atrial fibrillation: A report of the American College of Cardiology/American Heart Association Task Force on practice guidelines and the Heart Rhythm Society[END_REF]. The importance given to this supraventricular arrhythmia has been increased in the past few years and is expected to increase even more, since its mechanisms are not yet completely understood. This makes AF a challenging cardiac condition, considered as the last great frontier of cardiac electrophysiology. During AF, the P wave, that corresponds to a normal atrial activation, is replaced by f waves (fibrillatory waves), which are present through all the electrocardiogram (ECG) recording, but masked by the QRST complex that corresponds to the ventricular activity (VA) in each heartbeat.

To better understand the mechanisms behind AF, it is necessary to extract the atrial activity (AA) from the ECG for an accurate analysis of the f waves. A noninvasive method to extract the AA from the cardiac signals recorded by the standard 12-lead ECG is using matrix decompositions techniques to solve blind source separation (BSS) problems, such as principal component analysis (PCA) and independent component analysis (ICA) [START_REF] Rieta | Atrial activity extraction for atrial fibrillation analysis using blind source separation[END_REF]- [START_REF] Zarzoso | Extraction of ECG characteristics using source separation techniques: exploiting statistical independence and beyond[END_REF]. Matrix decomposition techniques have proved to be useful in AA extraction. However, it is known that they have some limitations since constraints need to be imposed to guarantee the uniqueness of such decompositions. Although mathematically coherent, such constraints may lack physiological grounds, making difficult the results interpretation.

To overcome the limitations of matrix decompositions, a tensor approach has been recently proposed to extract AA in AF ECGs [START_REF] Ribeiro | Atrial signal extraction in atrial fibrillation electrocardiograms using a tensor decomposition approach[END_REF]- [START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF]. Compared to matrix-based techniques, tensor decompositions present some remarkable features such as essential uniqueness with reduced or minimum constraints and the fact that the rank of the tensor can exceed its dimensions, whereas in matrices the rank is limited by the matrix lowest dimension. The block term decomposition (BTD) proposed in [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF] suits the characteristics of AA in an AF signal, since atrial signals can be approximated by all-pole (exponential) models and mapped onto Hankel matrices with rank equal to the number of poles [START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF]. The Hankel matrices containing the ECG data are stacked in the third dimension of a 3 rd -order tensor, and then processed by BTD. Previous experimental results in synthetic and real ECG data showed that BTD can outperform the matrix-based techniques for AA extraction in short [START_REF] Ribeiro | Atrial signal extraction in atrial fibrillation electrocardiograms using a tensor decomposition approach[END_REF]- [START_REF] De Oliveira | Source analysis and selection using block term decomposition in atrial fibrillation[END_REF] and long [START_REF] De Oliveira | Block term decomposition analysis in long segments of atrial fibrillation ECGs[END_REF] segments of AF ECG recordings.

The BTD has proved to be useful extracting the AA signal from the AF ECG, being able to outperform the matrixbased techniques. However, this tensorial approach was only assessed in short segments. Its performance in a whole ECG remains an open question. To answer this question and provide results about the temporal stability of this new promising technique, the present work assesses BTD for the first time in a whole ECG of a patient with persistent AF and compares it to two popular matrix-based methods for AA extraction: RobustICA-f [START_REF] Zarzoso | Robust independent component analysis by iterative maximization of the kurtosis contrast with algebraic optimal step size[END_REF] and PCA [START_REF] Jolliffe | Principal Component Analysis[END_REF]. Experimental results using Monte Carlo runs evaluate the intra-patient stability of BTD regarding the AA extraction from AF ECGs, showing the consistency of this technique.

The rest of this paper is organized as follows. In Section II, the notation used in the work is introduced. Section III recalls the BTD as a tensor approach to solve BSS problems and discusses quantitative measures of AA content. Section IV reports the experimental results and, finally, Section V presents the conclusion of this work and the perspectives of future works.

II. NOTATIONS

Scalars, vectors, matrices and tensors are represented by lower-case (a, b, c, ...), boldface lower-case (a, b, c, ...), boldface capital (A, B, C, ...) and calligraphic (A, B, C, ...) letters, respectively.

The transpose is represented by (•) T , symbol ||•|| represents the l 2 -norm and • represents the outer product. The operator diag(•) builds a diagonal matrix by placing its arguments along the diagonal. Given a 3 rd -order tensor A ∈ C I1×I2×I3 , with entries a i1,i2,i3 , its frontal slices are represented by A ..i3 ∈ C I1×I2 . Given a matrix A ∈ C I1×I2 , with entries a i1,i2 , its i th 1 row and i th 2 column are represented by a i1. and a .i2 , respectively.

III. METHODS

A. Block Term Decomposition

The BTD of an arbitrary third-order tensor T ∈ R I1×I2×I3 is written as:

T = R r=1 E r • c r (1)
where c r ∈ R I3 and E r ∈ R I1×I2 has rank L r and admits a decomposition E r = A r B T r , where A r ∈ R I1×Lr and B r ∈ R I2×Lr have rank L r . Equation ( 1) can be rewritten as:

T = R r=1 A r B T r • c r . (2)

If the matrix factors

A = A 1 A 2 . . . A R ∈ R I1× R r=1 Lr and B = B 1 B 2 . . . B R ∈ R I2× R r=1 Lr
are full-column rank, which requires that R r=1 L r ≤ I 1 , I 2 , and C = c 1 c 2 . . . c R ∈ R I3×R does not contain proportional columns, then the BTD is essentially unique [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF]Theorem 2.2].

The ECG data matrix, with K leads and N samples, can be modeled as:

Y = MS ∈ R K×N (3)
where M ∈ R K×R is the mixing matrix, modeling the propagation of the cardiac electrical sources from the heart to the body surface, S ∈ R R×N is the source matrix that contains the atrial, ventricular and noise sources, and R is the number of sources [START_REF] Zarzoso | Extraction of ECG characteristics using source separation techniques: exploiting statistical independence and beyond[END_REF]. The AA extraction in an AF ECG recording can be seen as a BSS problem, since the only data observed is the matrix Y, from which matrices M and S are to be estimated.

In [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF], the BTD is proposed as a solution of a BSS problem like [START_REF] Rieta | Atrial activity extraction for atrial fibrillation analysis using blind source separation[END_REF]. The idea to obtain a tensor from Y is to map its k th row onto a Hankel matrix and then stack each Hankel matrix along the third dimension (as frontal slices) of a third-order tensor Y ∈ R I×J×K , where

I = J = N +1 2 if N is odd or I = N 2 and J = N 2 + 1 if N is even.
This way, the tensor Y can be written as:

Y = R r=1 H (r) S • m .r . (4)
Equation [START_REF] Castells | Spatiotemporal blind source separation approach to atrial activity estimation in atrial tachyarrhythmias[END_REF] shows that the tensor ECG data follows a BTD tensor model. During AF, atrial sources can be represented by the exponential (or all-pole) model as:

s r,n = Lr =1 λ ,r z n-1 ,r (5)
where n = 1, ..., N , r = 1, .., R, L r is the number of exponential terms, z ,r is the th pole of the r th source, and λ ,r is the corresponding scaling coefficient. This way, their associated Hankel matrix accepts the Vandermonde decomposition as in [START_REF] Boley | Vandermonde factorization of a Hankel matrix[END_REF].

In the case of different poles, the Vandermonde matrix with L r ≤ I, J will have full-column rank L r , so if M does not have proportional columns, the BTD in (4) will be essentially unique. In the case of equal poles, milder conditions can assure the uniqueness of (4) [START_REF] Lathauwer | Blind separation of exponential polynomials and the decomposition of a tensor in rank-(lr, lr, 1) terms[END_REF].

B. Atrial Activity Content Measurement

Signal processing methods used to solve BSS problems separate the original signal in several sources. In AF ECG, typically at least one of these sources contains the AA. However, it is unknown if the AA is concentrated only in a single source. The present work, as previous ones, considers that the AA is concentrated in a single source, and the source with the most AA content is called the atrial source. A sucessful AA extraction depends on the accurate selection of the atrial signal among the others estimated sources. In [START_REF] De Oliveira | Source analysis and selection using block term decomposition in atrial fibrillation[END_REF] the BTD showed that the classical method of atrial source selection [START_REF] Rieta | Atrial activity extraction for atrial fibrillation analysis using blind source separation[END_REF], [START_REF] Castells | Spatiotemporal blind source separation approach to atrial activity estimation in atrial tachyarrhythmias[END_REF] does not provide a satisfactory accuracy, and then two sub-optimal methods that outperform the classical one were proposed. In this work the atrial source is selected using the method proposed in [START_REF] De Oliveira | Source analysis and selection using block term decomposition in atrial fibrillation[END_REF] that provided the highest accuracy.

In the frequency domain, the AA during AF has a peak between 3 and 9 Hz. The position of this peak is called dominant frequency (DF). We define as potential atrial sources any source with DF placed between such interval. To evaluate the AA extraction performance, three parameters are used to measure its quality. The first parameter is the spectral concentration (SC), that is, the relative amount of energy around the DF. The SC is computed as:

SC = 1.17fp 0.82fp P AA (f i ) Fs/2 0 P AA (f i ) (6)
where f p is the value of the DF, F s is the sampling frequency and P AA is the power spectrum of the AA signal, estimated as in [START_REF] Castells | Spatiotemporal blind source separation approach to atrial activity estimation in atrial tachyarrhythmias[END_REF]. The second parameter used to better analyze the potential atrial sources is the kurtosis, denoted K, of the signal in the frequency domain acquired by a 4096-point FFT. As in [START_REF] Zarzoso | Robust independent component analysis by iterative maximization of the kurtosis contrast with algebraic optimal step size[END_REF], the general expression of kurtosis is used, valid for noncircular complex data:

K = E[|s r. | 4 ] -2E[|s r. | 2 ] 2 -|E[s 2 r. ]| 2 E[|s r. | 2 ] 2 . ( 7
)
Kurtosis is a measure of peakedness and sparsity of a distribution. When computed in the frequency domain, it provides a quantitative measure of harmonicity of the signal. Indeed, a high kurtosis in the frequency domain means that the power spectral density is sparse, which is suggestive of a harmonic signal like AA in AF. Also, kurtosis is parameter free, not depending on the DF or the definition of a suitable interval for interpretation, as in SC [START_REF] De Oliveira | Source analysis and selection using block term decomposition in atrial fibrillation[END_REF]. The third parameter, which is used to eliminate weak sources, is the power contribution to lead V1 given by:

P (r) = 1 N ||m (V 1) r s r. || 2 (mV 2 ) (8)
where m

(V 1) r
is the contribution of the r th source to lead V1, given by the corresponding element of matrix M in Equation (3), and s r. is the r th source, corresponding to the r th row of matrix S, also in Equation ( 3). The power contribution to lead V1 by an AA source is expected to be strong, since this lead is the one that typically best reflects AA in AF ECGs.

IV. EXPERIMENTAL RESULTS

A. Real AF ECG Data and Preprocessing

The reported experiments consider 56 segments of a real standard 12-lead AF ECG recording from a patient suffering from persistent AF. These segments compose the whole ECG recording, that belongs to a database provided by the Cardiology Department of the Princess Grace Hospital Center, Monaco. The recording are acquired at a 977 Hz sampling rate and are preprocessed by a zero-phase forward-backward type-II Chebyshev bandpass filter with cutoff frequencies of 0.5 and 40 Hz, in order to suppress high-frequency noise and baseline wandering.

The first three segments in lead II from the patient is shown on Figure 1. All the segments have a fixed length of 1500 samples (about 1.53 seconds) and all the 12 leads are used in the experiments. The segments are downsampled by a factor of four, resulting in 3 rd -order tensors of dimensions 188 × 188 × 12, since the originally built 3 rd -order tensors of dimensions 750 × 751 × 12 pose some difficulties to Tensorlab.

B. BTD Setup

The BTD is implemented using the non-linear least squares (NLS) method available in Tensorlab MATLAB toolbox [START_REF] Vervliet | [END_REF] choosing R = 12 and L r = L, for r = 1, 2, ..., R, with L taking values in the set {17, 48, 95}. This choice is chosen based on the work [START_REF] Zarzoso | Parameter estimation in block term decomposition for noninvasive atrial fibrillation analysis[END_REF], that showed that such values provided good results for the heartbeat with the largest TQ segment of the patient considered in the present work. The tolerance threshold for BTD convergence is set to 10 -9 and the maximum number of iterations is set to 1000. Monte Carlo runs with Gaussian random initialization for the spacial and temporal factors at each run are used to analyze the performance of BTD in each segment regarding the AA signal extraction. Monte Carlo runs are needed since the performance of BTD depends strongly on the initialization of its factors and a suitable initialization is still an open challenge.

C. Experiments

Three experiments in the whole AF ECG recording are performed:

• BTD-1: A single Monte Carlo run for each segment using L r = 48. • BTD-2: 10 Monte Carlo runs for each segment using L r = 48, then the best performance out of the 10 independent runs is chosen, considering the quality of the estimation measured by the parameters introduced in the previous section. • BTD-3: For the segments with unsatisfactory AA extraction, i.e., with low values of SC and K, in the second experiment, 10 new Monte Carlo runs are performed changing the rank L r to 17 or 95 (the one which performed best), then the best performance out of the 10 Monte Carlo runs is chosen, as in the second experiment.

D. Intra-Patient Stability

In Figures 2 and4, the bluebox, the redline, the whiskers and the red dots represent the 25th and 75th percentiles, the median, the extreme values and the outliers, respectively, of the observed data.

Figure 2 shows how the SC of the atrial source is distributed over the 56 segments that compose the whole AF ECG for the three experiments performed with BTD, as well as PCA and RobustICA-f. The improvement of BTD-2 over BTD-1 is expected and illustrates the dependence NLS method on the initialization of the matrix factors, since the model parameters are kept fixed and only more Monte Carlo runs are performed. The improvement of BTD-3 over the previous experiments illustrates the need for choosing the right model parameters in order to have satisfactory performance. This improvement is also expected, since this experiment tries to adapt the multilinear rank to each segment, using the same number of Monte Carlo runs as BTD-2. matrix-based methods. These improvements can be viewed as the SC becomes more concentrated in high values, which means more stability.

In Figure 3, the temporal stability of BTD can be seen from another perspective, showing how the SC values of the AA signal varies over the 56 segments. For the sake of clarity, only BTD-3, the tensor version that provided the best performance, PCA and RobustICA-f are shown. It can be seen that the matrix-based methods present considerably variations over the whole ECG, whereas BTD-3 provides a SC more concentrated in high values, showing that this technique can be stable over time with a satisfactory AA extraction performance.

The superiority of BTD as an AA extraction tool over the matrix-based techniques is also shown in Table I. It can be seen that as the initialization of the factors and the model parameters are adapted to each segment, the average AA extraction quality is improved, as measured by SC and K. Since P (r) is a parameter used only to eliminate weak sources and not to measure the quality of the AA extraction, it is not Since the AA signals in AF ECGs are non-stationary, it is expected that the DF slowly changes with time, i.e., over the segments of the same recording. In Figure 4 it can be seen that the DF of the estimated atrial source by BTD-2, BTD-3, PCA and RobustICA-f lies in the interval 5.24-6.67 Hz, despite the outliers. In BTD-1 the DF interval is expected to be greater, since for many of the segments this method did not provide a satisfactory AA extraction, as we used fixed parameters to compute the BTD with a single initialization of its factors.

Indeed, BTD-1 provided an unsatisfactory AA extraction in 21 out of the 56 segments. For BTD-2 this number dropped to 5, while for BTD-3 no unsatisfactory AA extractions were performed. It is fair to report that PCA performed 9 unsatisfactory AA extractions and RobustICA-f only 7. In Figure 5 it can be seen that some estimated atrial sources for PCA, BTD-1 and BTD-2 have a P (r) value under 10 -4 mV 2 . This characterizes an unsatisfactory AA extraction, since a source with power contribution to lead V1 under this threshold does not present significant AA content, as reported in [START_REF] De Oliveira | Source analysis and selection using block term decomposition in atrial fibrillation[END_REF]. presents a strong dependence on its initialization and model parameters in this particular biomedical application. Nevertheless, with a suitable choice of its initialization and model parameters, BTD can provide not only good temporal stability, but a better estimation of the atrial source (as measured by SC and K) than two well-known matrix-based techniques. A method for computing BTD more robust to initialization is still an open challenge, as well as the optimal estimation of the model parameters for each AF ECG segment.

Taking this into account, future works would aim at developing a method of compute BTD with reduced dependence on initialization, as well as an automated method that can optimally estimate the model parameters for each particular observed segment. Increase the number of observed patients is also a prospect of future works, in order to analyze the inter-patient variability of this promising tensor technique.

Fig. 1 :

 1 Fig. 1: The first 3 segments of the AF ECG recording composed of 56 segments. Although all the 12 leads are processed, only lead II is shown for clarity.

Fig. 2 :

 2 Fig. 2: Distribution of SC values (%) for each version of BTD, as well as PCA and RobustICA-f.

Fig. 3 :

 3 Fig. 3: Variation of SC values (%) over the 56 segments for BTD-3, PCA and RobustICA-f.

Fig. 4 :

 4 Fig. 4: Variation of DF (Hz) for each version of BTD, as well as PCA and RobustICA-f.

  V. CONCLUSIONBTD has been recently proposed as an useful tool to atrial signal extraction in AF ECGs, showing satisfactory performances in fixed segments. The contribution of the present work is the analysis of the intra-patient variability of BTD. Three experiments show that the performance of this technique

Fig. 5 :

 5 Fig. 5: Power contribution to lead V1 values (mV 2 ) for the selected atrial source over the 56 segments for each version of BTD, as well as PCA and RobustICA-f.

TABLE I :

 I Mean values of SC (%) and K for the three versions of BTD, as well as PCA and RobustICA-f.

				SC	K
			BTD-1	53.02 151.33
			BTD-2	71.80 242.35
			BTD-3	76.95 260.41
			PCA	42.75 95.72
			RobustICA-f 66.10 216.06
	3	BTD-1	BTD-2	BTD-3	PCA	RobustICA-f

ACKNOWLEDGMENT Pedro Marinho R. de Oliveira is funded by a PhD scholarship from the IT Doctoral School of the Université Côte d'Azur. Vicente Zarzoso is a member of the Institut Universitaire de France.


domain: spi info

text:
Cuong Cong

Nada Pham

Alexandre Matta

Benoit Durupt

Marianne Eynard

Guillaume Allanic
email: mallanic@cadesis.com

Marc Ducellier

Philippe Joliot

Boutinaud
email: pboutinaud@cadesis.com

Cong Cuong Pham
email: phamcong@utc.fr

Nada Matta
email: nada.matta@utt.fr

Alexandre Durupt
email: alexandre.durupt@utc.fr

Benoit Eynard
email: benoit.eynard@utc.fr

Marianne Allanic

Guillaume Ducellier
email: guillaume.ducellier@utt.fr

Marc Joliot
email: marc.joliot@u-bordeaux2.fr

Philippe Boutinaud

Using Ontologies to Access Complex Data: Applications on Bio-Imaging

Keywords: Knowledge sharing, Data Management, Bio-Imaging, PLM

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Information Systems, used to share information, lead to the growth of heterogeneous data and then the dependencies between them. Thus, the links and dependencies among heterogeneous and distributed data are more and more complex during daily activities of users (researchers, engineers, etc.). The data exploitation (interrogation and sharing) has to be adapted to the context of large data and complex dependencies. To overcome this current inconvenience, more and more research works have investigated and studied the Semantic Web (SW) concepts and techniques to give a cognitive access to information and data. Our contribution is to propose a methodology to facilitate the exploitation (interrogation and sharing) of data in an organization. Techniques to handle heterogeneous and complex data structuring are defined in order to support the dynamic evolution of this type of data. A request system using ontology and semantic mechanism is also developed in order to offer a user friendly data management system. This work is defined for Bio-Imaging domain.

Related Work

Data access or querying data is the data search process of users to answer a specific question. It is an important function of any information system. From very first development of database technologies, querying data has been a function often dedicated to Information and Technology (IT) experts. Providing non-IT people like end-users with an efficient way to query database, semantic query for instance, was always a challenged topic. In current BMI data management system, some methodologies have been proposed to enable the data access of end-users.

Riazanov et al, [START_REF] Riazanov | Semantic Querying of Relational Data for Clinical Intelligence: A Semantic Web Services-Based Approach[END_REF] proposed a semantic querying of relational data for clinical intelligence. For the authors, "self-service ad-hoc querying of clinical is problematic as it requires specialized technical skills and the knowledge of particular data schemas". A semantic querying allows end-users (clinical researchers, surveillance practitioners, health care managers…) to formulate queries in terms of domain ontologies which are more understandable than data schemas. User queries are then transformed to the one on the data sources by using a mapping between ontologies' terms and data sources.

LORIS [START_REF] Das | LORIS: A Web-Based Data Management System for Multi-Center Studies[END_REF] is a web-based data management system for multi-center studies from data acquisition to processing and dissemination. The main querying function of LORIS is Data Querying Gui (DQG). By using a web-based interface, DQG "allows researchers to design, execute, and save queries in a simple and intuitive manner, without having to write complex SQL queries". Ping et al. [START_REF] Ping | A Web-Based Data-Querying Tool Based on Ontology-Driven Methodology and Flowchart-Based Model[END_REF] also developed a web-based dataquerying tool using ontology-driven methodology and flowchart-based model. The former was used to formulate the query task through a Protégé plugin, the later executes queries and presents the result through a visual and graphical interface.

In the BIRN [START_REF] Helmer | Enabling Collaborative Research Using the Biomedical Informatics Research Network (BIRN)[END_REF], BIRN Mediator helps users make easily a query to a collection of data sources (such as relational or XML databases, or web services) by provide them with a single consistent virtual schema while the actual data resides in remote sources under their original schemas. "The mediator maps the sources schemas to the common domain schema, using declarative logical formulas, transparently to the user. The mediator provides the most recent data available, since the user queries are translated into source queries and executed at the sources in real-time".

All these proposed methodologies have been based on the well-known approach called Ontology-Based Data Access (OBDA). This approach uses ontology, the pivot component of the Semantic Web (https://www.w3.org/standards/semanticweb/) to enable the semantic data access of end-users by provide them with a semantic representation of data. In this chapter, we propose to use this techniques in order to support knowledge sharing in Bio-Imaging at Gin Lab.

Knowledge sharing in Bio-Imaging

Several researchers in Bio-Imaging need to share their results and data they use. They manipulate heterogeneous data: human information, brain images, diseases descriptions files. They produce more images, statistics data, diseases and brain descriptions. Information grown very quickly and they need a support that handle data evolving than classic database cannot tackle.

Bio-Imaging researchers in GIN lab has been interviewed in order to identify their real needs and their difficulties to share and use information using existing information system during daily activities. Most of scientists have difficulties in data querying and they almost cannot accomplish this task without helps of database technicians. In fact, their data base in very complex; there is a lot of links between data and it is not very easy for them to discover the data base structure in order to understand where a data can be inserted. Otherwise, the data names in the database do not reflect their use. So, when they want to obtain information from data base, they describe their needs, and a technician try to answer them. The problem is on the interaction with database. They need to use incremental requests in order to discover existing data and apply some process on it. Sometimes, they repeat process techniques already done by other researcher; they need to discover their colleague results and use them. Finally, technician cannot update the Database because of its complexity.

Currently, they extract a copy from data base in Excel files (example Fig 1 ) and each researcher deal with his excel file. The problem is each one, use specific name of the same data. Results are not shared and used. There is a duplication of the database and its structure. As conclusion, there is no information sharing between researchers due of the complexity and the dynamic updating of database.

Example of data Excel File

To answer these problems, we propose:

• To organize data in incremental way, basing on the main object. In the Bio-imaging case, it is the studied person (named as subject). So data identified and produced concerns this subject.

• To use a database management system that enable easily to manage and update this type of data organization. For instance, in our application, we use Product Life Cycle Management Tool (PLM) in order to handle this type of organization. In fact, structure of data in PLM is organized as the evolution of the product from the idea to the concept and product characteristics. Adding of that, several forms of data can be supported : data, images, text files, etc. We know that different data management systems exists (SQL, NoSQL, etc.) and data organizations (object, relational, etc.) [START_REF] Ramakrishnan | Database management systems[END_REF]. The PLM system, we used is based on relational data base. We prefer to use a PLM system in which the organization of data correspond to the need of Gin lab bioimaging researchers; keeping data as an evolution of studies around a subject.

• To develop a request interface as a support of domain vocabulary and links to database. So, a domain ontology has been defined and a request system has been developed in order to handle knowledge sharing between bio-imaging researchers at Gin-lab.

Structuring data

As mentioned above, complex data will be structured as relations of a main object. In Bio-imaging studies, the subject (person who follows a medical protocol) is central. So data are structured as to subject as follows:

• subject: number, name, birthdate • exam: investigator, type, tool, protocole, results

• process: investigator, type, tool, results, references

To support the dynamic evolution of data, PLM TeamCenter tool are used.

Product Lifecycle Management

The Product Lifecycle Management (PLM) systems integrate constantly all the information produced throughout all phases of a product's lifecycle to everyone in an organization at every level (managerial, technical…) [START_REF] Sudarsan | A product information modeling framework for product lifecycle management[END_REF]. This type of tools are developed to help product designers and to provide a traceability support of the evolution of n artefact from the requirements to the product use and even recycling (Fig 3). We can figure some key advantages of PLM systems [START_REF] Eynard | UML based specifications of PDM product structure and workflow[END_REF]:

• Establishing an effective PLM system reduces the enormous data resources to a coherent data flow, avoids redundancies and heterogeneities. • PLM enables the collaboration through distributed and virtual/extended enterprises (workflow and process management, communication and notifications, secure data exchange...) • PLM permits the product structure and its evolution management during different steps and track-performed modifications tracking. • PLM is a mature solution to tackle the heterogeneity, growth and complexity of the data and its processing methods as well as some of the traceability and confidentiality issues. So, PLM system brings together: Products, service, activities, processes, people, skills, data, knowledge, procedures and standards. It provides an efficient solution to handle the complex and heterogeneous data resources and a mature method to track the evolution and modification of these data. However, along with these advantages, it also exists some issues:

• Lack of strong stakeholders, ICT tools as well as a common standard between PLM systems causes data integrity problems and limits the access to and sharing of product information and knowledge distributed, • Another issue of PLM community is the increasing of need for product lifecycle knowledge capitalization and reuse in order to reduce time and cost. • Database exploitation requires a good understanding of database structure as well as data model especially in the context where the data is heterogeneous and the links and dependencies among data are complex.

Data Representation in TeamCenter

Classification in corresponding with the BMI-LM data model

In this classification, the nature of data is repeated. In fact, we have images classes as results and as entry data. It is the same for processes, descriptions, etc. The low-level expression of UML schema and the complex relations among classes in the classification also brings difficulties for users in querying the database. To overcome this issue, we build an ontology, which bases on both of data model and classification. This ontology shows logic representation of information in Bio-Imaging and it provides an overview of concepts in the data model and the relationship among them but now represented in a natural language, and therefore it allows end-users to create a query close to his reasoning.

Data access using ontology

The concept « ontology » has been used a long time ago in different communities. In the area of computer sciences, "an ontology is a special kind of information object or computational artifact" [START_REF] Guarino | What Is an Ontology?[END_REF]. In 1993, Gruber [START_REF] Gruber | A Translation Approach to Portable Ontology Specifications[END_REF] defined an ontology as an "explicit specification of a conceptualization" while four years later Borst [START_REF] Borst | Construction of Engineering Ontologies for Knowledge Sharing and Reuse[END_REF] defined it as a "formal specification of a shared conceptualization". This definition implied that the conceptualization should be readable by machine (formal format) and should be expressed a shared view between several parties, that means a consensus rather than an individual view. Merging Gruber's and Borst's definitions, Studer [START_REF] Studer | Knowledge Engineering: Principles and Methods[END_REF] stated that: "An ontology as a formal explicit specification of a shared conceptualization" (Erreur ! Nous n'avons pas trouvé la source du renvoi.). The use of ontology brings some benefits:

• Support communication and cooperation among systems: Ontology enables interoperability, and integration of heterogeneous data sources. • Enable the knowledge sharing and knowledge reuse.

• Enable content-based access and provide automated services based on machine (the key component of the Semantic Web). Data access using ontology or Ontology-based data access (OBDA) is a new paradigm for data integration and accessing data sources with complex structure [START_REF] Khamis | Modified Query-Roles Based Access Control Model (Q-RBAC) for Interactive Access of Ontology Data[END_REF] It aims to provide end-users with a semantic access to databases by using a three-level architecture [START_REF] Lenzerini | Ontology-Based Data Management[END_REF] containing:

• Conceptual layer (ontology layer),

• The data sources,

• The mapping between the ontology and the data source.

The ontology acts as a mediator between user and the data source. Its aims to provide user with a semantic representation of data sources by using a set of concepts in the domain of interest and all relations among them. The data sources are the repositories of data stocked in a relational or non-relational databases. The mapping layer maps the domain concepts to the data sources.

The queries formulated by users using concepts of ontology are translated to the one on the data sources by using this explicit mapping (Fig 7).

Architecture of OBDA systems

This principle are used to define an Ontology Based System in order to support data acces in Bio-Imaging at Gin lab. Before describing the architecture of this system, letus define the used ontology.

Bio-Imaging Ontology definition

There is a lot of ontology defined in medicine, for instance, in oncology, neurology [START_REF] Gibaud | OntoVIP: An ontology for the annotation of object models used for medical image simulation[END_REF].

But, little work study bio-imaging representation. Gibaud et al, [START_REF] Gibaud | NeuroLOG: sharing neuroimaging data using an ontologybased federated approach[END_REF] define concepts used in Bio-imaging like: Dataset, Processing, Investigators, Medical Image files, Equipment, subjects, etc. (Fig 8). Conceptual tree of GIN lab ontology

Ontology -Data link

Ontologies' low-level concepts have to be linked to data in database, or unless respect the variable name of these data. An inference engine can help to build a data request and generate links using the propagation of relations between concepts (Fig 11).

To generate request on database, relations between concepts and data types are defined.

For each concept, several data types are linked. For instance, acquisition protocol is related to different protocols in magnetic-resonance-imaging acquisition process.

These links are integrated in XML file as interface between Inference engine and database management tool (the TeamCenter in GIN lab).

A request system is then developed in order to provide a user friendly data request system. Links between Ontology and database model.

Ontology and classification mapping.

A table of mapping between ontology concepts and classes is required since only classes of the classification of data are connected to data. Using this table of mapping, the query formulated by users with vocabularies of ontology will be translated to the one that is understandable and executable by a Query Processor.

One concept can be connected with many classes and vice versa. To simple the mapping, firstly we used some visual and interactive tools (Free-mindhttp://freemind.sourceforge.net for instance) to map classes to concepts. The mapping was then transformed into XML and integrated in the query system. F i g 1 2 illustrates the mapping between concept "image-acquisition-protocol" of imaging ontology (Fig 12Erreur ! Nous n'avons pas trouvé la source du renvoi.) and all leaf classes in branch "AcquisitionDefinition-Branch/ Imaging" of the classification of data.

Ontology based query system

By using ontology tree and ontology graph, ontology-based graph query interface helps users to make a query more easily. Using the ontology tree and ontology graph, users can understand the relationships among concepts and directly choose query parameters.

Users also can choose a query in query history to re-execute, modify or complete it. When a user completes his query, our system does as following:

1. Identifying nodes links, following relations in ontologies.

2. Generating an output query in a format understandable and executable by Query Processor, XQuery Engine for example. With the support of these graph, user understand relationships among data in the database, he/ she defines the conditions of query according to their purpose. We take here an example of query frequently used by scientists at GIN lab: "Querying about Brain image results from Retin Treatment using Dash protocol corresponding to men less than 45 years old, left handled and passed exam before January 2013"

If a researcher tries to define a SQL query using PLM related to this question, without using ontology. He/she has to know relations between Study Subject, Exam results, Processing definition and Processing Results data in PLM Database. First, the name of objects in data base are more computer driven called and he has to follow all links in order to identify the ones corresponding to his query which can then be:

Select Image Results from Processing Results

Where Processing Protocol = "Dash" and Processing Treatment = "Retin" and from Acquisition Result where Acquisition Date < "January 2013"and from Study Subject where Study Subject Gender = "men", Study Subject Handless = "Left handled" Study Subject Age <"45".

Using Ontology Interface (Fig 13), she/he can select start by select the concept "Subject", as for concept parameters, selects "gender" and "Age", writes her/his choice (Men and <45 years old). Then she/he selects the concept "Protocol", writes the choice (= "Dash"), selects the concept "Treatment Process", choice (= "Retin"), and "Treatment results" (File = "Image"), she/he selects concept "Acquisition result" (Acquisition psychology laterality handless = "Left handled") and (Date of Acquisition < "January 2015). The inference engine identifies relations between Subject, Acquisition and Process:

• Acquisition "Use" Source

• Protocol "Use" Acquisition • Treatment "Follow" Protocol • Treatment "generate" Treatment Result By subsuming, relations are inherited so, the result of the inference engine will be: Treatment-Result-Type (=Image) "Follows" Protocol-Name (=Dash) and Protocol-Treatment (=Retin) "Uses" Acquisition-Laterality-Handless-result (=Left handled) "Uses" Subject-Gender (=Men) and "Uses" Subject-Age (<45).

This result query description is then transformed on a TCquery that considers PLM database as a XML file, the output query is transformed to xQuery in order to extract data from that file: declare function local:getQueryObject(){ let $query := <query query_name='Item...'> <param name='Type' value='ProcessRes'/> </query> return Teamcenter:Query($teamcenter, $query) }; let $processingResults := local:getQueryObject() for $processingResult in $processingResults return if(fn:count($processingResult[(@protocol = "Dash") and (@treatment = "Retin")]/F_GIN4_rel_Acquisition/GIN4 _Acquisition[@laterality_handless = "Left handled" ]/B_ GIN4_rel_ExamenRes/GIN4_ExamRes[@examDate <= "01 January 2013"]/B_GIN4_rel_ExamRes/GIN4_StudySub [(@ gin4_sex = "M") and (@gin4_databirth >= "1970")]) >0) then (processingResult) else ()

The results of the query will be then returned and represented in a graph or a table in the same query interface. Users can visualize data to understand the relationships among them or to find hidden information and inferred knowledge.

CONCLUSION

In this complex and heterogeneous data management problem is studied; How to search data without knowing the data structure, how to discover data defined by actors in an organization, how to share knowledge and data in friendly way. A general approach for ontological model construction and an ontology-base query interface is presented as a solution to tackle the difficulties in querying complex database. Data will be organized around objects. For that, specific data organization systems can be used as PLM. PLM help to organize data as the evolution description of product parts. an architecture and request system are developed in order to build links between data management and semantic system. A use case in Bio-Imaging domain has been also used to illustrate the abilities of our proposed interface.

As future work we will focus on the test of proposed query interface with various queries sets (in Bio-Imaging domain) and engineering design (in PLM). The ontology tree and ontology graph must be also developed to cover all concepts in Bio-Imaging domain. Ontology will be implemented in semantic web language (RDF, SPARQL) in order to use inference engine for information search. We propose to generate an alert system when new data are added with semantic annotation of these data in order to show the reason of their creation.

  Fig 1.Example of data Excel File

Fig 2 Fig 2 .

 22 Fig 2 illustrates the needs of knowledge sharing at Gin-Lab

Fig 3 .

 3 Fig 3.PLM Functions.

Fig 4

 4 Fig 4 presents the BMI-LM (Bio Medical Imaging Life cycle Management) data model used in the PLM "TeamCenter 9.1" [1]. By adopting PLM solutions in the context of Bio-Imaging, this PLM-oriented data model covered the whole stages of a BMI study from specifications to publications and enabled the flexibility in data management.

Fig 4 .

 4 Fig 4. BMI-LM data model implemented in Teamcenter 9.1 BMI-LM contains three types of objects: Result objects (Exam, Acquisition, Data Unit, Processing), Definition objects (Exam, Acquisition, Data Unit, Processing) and Reference objects (Bibliographical, Data). "Definition" concepts have been used in order to enable the reuse of data. For example, all the Processing results computed by using the same Acquisition device and the Processing parameter can be attached to the same corresponding Processing definition. The classification (Fig 5) has been built based on the data model. From that, BMI data have been classified into branch, classes and subclasses. The classification allows a specific class to be added to a generic item (object in the data model).In comparing with the data model, the classification and its attributes are easier to modify for user than objects attributes, it is good to fit the model flexibility requirement and also for the appropriation of the database by user[START_REF] Allanic | Application of PLM for Bio-Medical Imaging in Neuroscience[END_REF].

Fig 5 .

 5 Fig 5.

Fig 6 .

 6 Fig 6.Definition of ontology[START_REF] Studer | Knowledge Engineering: Principles and Methods[END_REF].

Fig 7 .

 7 Fig 7.

Fig 8 .

 8 Fig 8.ontology for Bio-Imaging.

Fig 9 .

 9 Fig 9.

Fig 10 .

 10 Fig 10. Conceptual graph of GIN lab ontology

Fig 11 .

 11 Fig 11. Architecture of Knowledge sharing in PLM

Fig 12 .

 12 Fig 12. Ontology concepts tree and the classification of data mapping.

3 .

 3 Executing the output query on PLM data file (.xml or. json format). 4. Results are then visualized as a graph and data in the Interface query.

Fig 13 .

 13 Fig 13. Ontology-based graph query interface

  Fig 14 presents an extract of results represented in a graph.

Fig 14 .

 14 Fig 14. Graphical representation of results. Click on a node to highlight all related nodes.

We adapt this ontology corresponding of the logic use of information in database. Information belongs to three major categories: Tools, Data, and Process (Fig 9).

Several relations exist between these concepts like Use, Follow and Provide (Fig 10).

Acknowledgments. The work presented in the paper is conducted within the ANR (Agence Nationale de la Recherche) founded project BIOMIST.