P Abry

J Spilka

R Leonarduzzi

V Chudáček

N Pustelnik

M Doret

Sparse learning for intrapartum fetal heart rate analysis

Keywords: Intrapartum Fetal Heart Rate, Acidosis prediction, labor Stage, Sparse learning, scale-free, learning generalization

Fetal Heart Rate (FHR) monitoring is used during delivery for fetal well-being assessment. Classically based on the visual evaluation of FIGO criteria, FHR characterization remains a challenging task that continuously receives intensive research efforts. Intrapartum FHR analysis is further complicated by the two different stages of labor (dilation and active pushing). Research works aimed at devising automated acidosis prediction procedures are either based on designing new advanced signal processing analyses or on efficiently combining a large number of features proposed in the literature. Such multi-feature procedures either rely on a prior feature selection step or end up with decision rules involving long lists of features. This manyfeature outcome rule does not permit to easily interpret the decision and is hence not well-suited for clinical practice. Machine-learning-based decision-rule assessment is often impaired by the use of different, proprietary and small databases, preventing meaningful comparisons of results reported in the literature. Here, sparse learning is promoted as a way to perform jointly feature selection and acidosis prediction, hence producing an optimal decision rule based on as few features as possible. Making use of a set of 20 features (gathering "FIGO-like" features, classical spectral features and recently proposed scale-free features), applied to two large-size (respectively 1800 and 500 subjects), well-documented databases, collected independently in French and Czech hospitals, the benefits of sparse learning are quantified in terms of: i) accounting for class imbalance (few acidotic subjects), ii) producing simple and interpretable decision rules, iii) evidences for differences between the temporal dynamics of active pushing and dilation stages, and iv) of validity/generalizability of decision rules learned on one database and applied to the other one.

Introduction

Intrapartum fetal monitoring. Worldwide, obstetricians routinely monitor, during delivery, fetal heart rate (FHR) to detect oxygen deprivation in fetuses. Their main objective is to predict fetal acidosis as early as possible, so as to make timely decisions for operative deliveries that prevent adverse outcomes (neural development disability, neonatal encephalopathy, and cerebral palsy) (Chandraharan & Arulkumaran 2007). Clinical routine for FHR evaluation remains based on visual inspection of FHR, relying on guidelines issued by the International Federation of Gynecology and Obstetrics (FIGO) (FIGO 1986, Ayres-de Campos et al. 2015) mostly focusing on decelerations (number, shape, depth and duration), variability, and baseline levels. This practice, however, is well known to suffer from substantial inter-and intra-observer variability [START_REF] Hruban | [END_REF], Blackwell et al. 2011[START_REF] Spilka | [END_REF] and to induce unnecessary intervention or operative deliveries, increasing uselessly cesarean section rate, that are a posteriori found to have been avoidable (False Positive) (Alfirevic et al. 2006). Improving FHR analysis by having recourse to advanced statistical signal processing tools has thus been the topic of on-going research efforts. Related works: feature design. Beyond attempts to automatize the computation of FIGO features-or generalized versions (cf., e.g., (Parer et al. 2006, Spilka et al. 2012[START_REF] Nunes | [END_REF]))-frequency-based features were used [START_REF] Siira | [END_REF], following standard approaches for adult heart rate variability analysis (Akselrod et al. 1981). Such features are often referred to as linear since they essentially quantify autocorrelations in FHR (Gonçalves et al. 2006, Laar et al. 2008[START_REF] Magenes | Proc. 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society[END_REF], Siira et al. 2013). Further, advanced statistical signal processing was involved in FHR analysis. Notably, complexities in temporal dynamics were quantified using information theoretic quantities, such as entropy rates (Costa et al. 2002, Echeverria et al. 2004, Porta et al. 2013[START_REF] Spilka | Annual International Conference of the IEEE[END_REF], Granero-Belinchon et al. 2017), various nonlinear transforms [START_REF] Magenes | Proc. IEEE-INNS-ENNS International Joint Conference on Neural Networks IJCNN[END_REF][START_REF] Magenes | Proc. 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society[END_REF], Chudáček, Anden, Mallat, Abry & Doret 2014, Georgieva et al. 2014), or the scalefree or (multi)fractal paradigm (Francis et al. 2002, Doret, Helgason, Abry, Gonçalvès, Gharib & Gaucherand 2011[START_REF] Abry | 26th International Symposium on Computer-Based Medical Systems (CBMS)[END_REF], Doret et al. 2015). Such features are termed nonlinear as they probe the dependencies in FHR temporal dynamics potentially beyond mere autocorrelations. For overviews, see e.g. (Spilka et al. 2012, Haritopoulos et al. 2016). Related works: feature selection and machine learning. Faced with the modest acidosis-prediction performance achieved by standalone features, numerous works combined several features through supervised machine learning strategies (cf., e.g., (Bernardes et al. 1991, Costa et al. 2009[START_REF] Warrick | [END_REF], Georgieva et al. 2013, Spilka et al. 2012, Czabanski et al. 2012[START_REF] Warrick | [END_REF], Xu et al. 2014, Frasch et al. 2014, Dash et al. 2014, Spilka et al. 2017)). Most of these contributions acknowledge several problematic issues. First, the use of small-size databases (from tens to hundreds of subjects) and of a large number of features results in a lack of robustness and generalizability (Frasch et al. 2014). Second, complicated decision rules involving too many features are not suited to clinical practice as clinicians often need to conceptualize what their decision is based on (Spilka et al. 2012, Xu et al. 2014)-feature selection is thus often performed prior to learning, but without guarantees on the joint optimality of both procedures. Third, the use on one database of decision rules learned from another does not always yield satisfactory performance. Related works: labor stages. FHR analysis is further complicated by the existence of two distinct stages during labor. The dilation stage (stage I) consists in progressive cervical dilation and regular contractions. The active-pushing stage (stage II) portrays a fully dilated cervix combined with maternal pushing efforts. Analyses have been performed either globally, mixing both stages (Costa et al. 2009[START_REF] Warrick | [END_REF], or focusing on stage I only (Spilka et al. 2017). Differences in the dynamics of each stage remain barely documented (see a contrario (Spilka, Abry, Goncalves & Doret 2014, Lim et al. 2014, Spilka et al. 2016b, Granero-Belinchon et al. 2017)). Outline, goals, and contributions. The present contribution aims to assess the potential benefits, for acidosis prediction, of sparse learning-i.e. the combination of feature selection and classification into a single procedure. It also aims to show that decision rules learned for each stage need to be different and based on different features. Finally, it aims to validate that sparse learning yields decision rules that satisfactorily generalize-i.e. they remain valid on samples not used for training. To that end, the Sparse Support Vector Machine (S-SVM)-detailed in Section 3-is applied to FHR data from two independent large-size databases-described in Section 2. Results, discussed in Section 4, highlight the benefits of sparse classification rules, and enhance the evidence for differences between the temporal dynamics of both stages. Further, the generalization ability of S-SVM is explored through a cross-database evaluation.

The present work complements and strengthens preliminary attempts reported in (Spilka et al. 2016a, Spilka et al. 2016b, Spilka et al. 2017).

Data: databases and datasets

Databases

Two independent large-size databases are used in the present work. They were collected-with different technologies and constraints-at academic hospitals in Lyon, France (LDB), and Brno, Czech Republic (BDB). They share comparable clinical characteristics in terms of gestational age larger than 36 weeks, cf. Table 1 for details.

For LDB, FHR data were collected at the French public Hospital Femme-Mère-Enfant, from 2000 to 2010, during routine labor monitoring. The acquisition was performed using STAN S21 or S31 devices (Neoventa Medical, Molndal, Sweden) via internal fetal scalp electrodes (12-bit resolution and sampling rate 500 Hz). LDB contains 3049 recordings, and is documented with relevant clinical information such as (Chudáček, Spilka, Burša, Janků, Hruban, Huptych & Lhotská 2014).

Tracings were included in the present study according to clinical and data-quality criteria (Doret, Massoud, Constans & Gaucherand 2011, Spilka et al. 2017): gestational age ≥ 37 weeks, maternal age ≥ 18, tracing ending less than 20 minutes before delivery, after-delivery pH measurement available, less than 50% of missing data in either stage of the delivery process.

Datasets

For each database, tracings are split into two datasets, depending on the duration of the second stage (t II ): Set S I is defined as t II ≤ 15 min and corresponds to births during stage I or in the early phase of the stage II. Set S II corresponds to t II > 15 min and thus to births during (a possibly long) stage II.

Assessment of the newborn's health status at delivery is based on pH, measured from an immediate post-birth umbilical-cord-artery blood sample, and used as a signature of fetal acidosis in the minutes before delivery (Amer-Wåhlin et al. 2001). Subjects were hence split into two classes: acidotic defined as pH ≤ 7.05 and normal with pH > 7.05. Clinical information on the resulting datastets is summarized in Table 1.

Data preprocessing, FHR time series, and analysis

Collected tracings consist of a list of R-Peak intervals (RRi) in ms. They were first corrected for outliers and missing data by a sliding median filter. RRi lists were resampled using linear spline interpolation to yield regularly-sampled beat-per-minute (bpm) time series X(t). Sampling frequencies were set to f s = 10 Hz for LDB and f s = 4 Hz for BDB-due to the different sampling frequencies in both databases.

Because pH measured at delivery can only be reminiscent of the health status of fetuses in the last minutes of labor, FHR analysis was only conducted in the last 20 min of the first stage for S I , and the last 20 min before delivery for S II , as sketched in Fig. 1.

X NI (t) . . . X 2 (t) X 1 (t) ≤ 15 min. stage II stage I I → II X N2 (t) . . . X 2 (t) X 1 (t)
> 15 min. 3. Methods: Feature Design and Sparse Learning

stage II stage I I → II

Feature Design

Following preliminary works reported in (Spilka et al. 2016a, Spilka et al. 2016b, Spilka et al. 2017), a set of 20 features is used, chosen amongst features either widely used in the literature (such as automated FIGO-like features), derived from adult heart rate analysis (such as spectral features), or more recently shown to be relevant for intrapartum fetal heart rate analysis (such as scale-free features). Table A1 in Appendix A summarizes the feature lists as well as other acronyms used throughout the manuscript.

3.1.1. Automated FIGO-like features (9). Nine FIGO-inspired features are used: (β 0 , β 1 , LT V, ST V, #acc, #dec, MAD dt , T stress , A dec ), as devised in (Chudáček et al. 2011, Spilka et al. 2012). They quantify: baseline B(t) level and evolution as B(t) = β 0 + β 1 t; long-and short-term variabilities (LTV, STV) (Ayres-de Campos et al. 2015); accelerations/decelerations (Ayres-de Campos et al. 2015), via their numbers (#acc and #dec), average depth (MAD dt ), average duration (T stress ) and average area (A dec ).

Spectral features (5)

. Spectral estimation is conducted over using the Welch periodogram [START_REF] Manolakis | Statistical and Adaptive Signal Processing[END_REF]. For adults, energies in well-defined frequency bands are known to be associated with the sympathetic/parasympathetic balance. Because no widely-accepted equivalent definitions are available for fetuses (Siira et al. 2013, Doret et al. 2015), the same bands as those for adults are used: very low frequency E V LF ([0.003, 0.04] Hz), low frequency E LF ([0.04, 0.15] Hz), and high frequency E HF ([0.15, 0.40] Hz). Further, the ratio LF/HF of E LF and E HF , and the spectral index α (estimated over LF and HF ), are computed.

3.1.3. Scale-free features (6). Scale-free and multifractal features were recently shown to offer relevant and robust alternatives to the classical measurements of long-and short-term variabilities (STV and LTV), cf. (Doret, Helgason, Abry, Gonçalvès, Gharib & Gaucherand 2011[START_REF] Abry | 26th International Symposium on Computer-Based Medical Systems (CBMS)[END_REF], Doret et al. 2015). The Hurst parameter H (cf., e.g, [START_REF] Samorodnitsky | Stable non-Gaussian random processes Chapman and Hall[END_REF] for a definition) is a linear feature, as it describes the autocorrelation function or the Fourier spectrum, yet in a scale-free spirit [START_REF] Abry | 26th International Symposium on Computer-Based Medical Systems (CBMS)[END_REF], Doret et al. 2015). It has been shown that it can be efficiently and robustly computed from discrete wavelet transforms [START_REF] Abry | [END_REF], Abry & Didier 2018). Multifractal parameters (h min , c 1 , c 2 , c 3 , c 4 ) (cf., e.g., (Wendt et al. 2007) for a definition) produce an advanced scale-free characterization of a time series' variability. Indeed, they describe the fluctuations of its regularity along time, based on the full dependence structure-not just the autocovariance. Efficient and robust estimators of multifractal parameters are obtained from wavelet leaders, a nonlinear transform of wavelet coefficients (Wendt et al. 2007). Parameters h min and c 1 remain mostly driven by the autocorrelation and are hence closely related to H. Conversely, parameters (c 2 , c 3 , c 4 ) convey information not encoded in the autocorrelation and thus complement H, c 1 , h min . They are hence nonlinear features, quantifying the variability, asymmetry and heavy-tailness in FHR's temporal dynamics (Wendt et al. 2007, Doret, Helgason, Abry, Gonçalvès, Gharib & Gaucherand 2011).

Sparse Learning

3.2.1. Support Vector Machine. A Support Vector Machine (SVM) is a classical machine learning procedure [START_REF] Hastie | The Elements of Statistical Learning Springer Series in Statistics[END_REF], already used for acidosis classification, cf., e.g., [START_REF] Warrick | [END_REF], Xu et al. 2014). Let x n ∈ R P denote P -dimensional feature vectors for each of the N subjects, and let y n ∈ {-1, 1} be the corresponding class labels. SVM produces a decision rule d = sgn(w T x n + b) to classify the subjects. The parameters (w, b) are estimated by minimizing a functional that enforces classification performance through the hinge loss function

F w,b (x, y) = max 0, 1 -y(w T x + b) : ( w, b) ∈ argmin w∈R P , b∈R 1 2 w 2 2 + C N n=1 F w,b (x n , y n ), (1)
where the regularization hyper-parameter C > 0 controls data sparsity. Indeed, when solving Eq. ( 1), only a few samples (the so-called support vectors, whose overall amount is indirectly induced by C) actually contribute to the design of the decision rule. While (1) can be easily solved by gradient descent algorithms, SVM suffers from several documented limitations (cf., e.g., [START_REF] Warrick | [END_REF])): i) data sparsity is not well suited to biomedical applications, where feature distributions are often largely intertwined for the two classes; ii) decision rules involve essentially all proposed features-w p = 0, ∀p-thus making their interpretation difficult (a severe impairment for transfer towards clinical practice) ; iii) While SVM show some robustness to unbalanced class sizes thanks to potential support vector sparsity, the special context of fetal acidosis prediction is characterized with extremely unbalanced sizes of the acidotic and healthy classes, N + /N - 1, which requires special attention.

3.2.2. Sparse Support Vector Machine. To overcome such issues, feature selection procedures were envisaged (Xu et al. 2014, Bron et al. 2015[START_REF] Soguero-Ruiz | Biomedical and Health Informatics[END_REF]).

Yet, the concatenation of two phases (feature selection and classification) casts shadows on their joint optimality. Instead, Sparse Support Vector Machines (S-SVM) aim to perform both operations jointly [START_REF] Hastie | The Elements of Statistical Learning Springer Series in Statistics[END_REF], Bach et al. 2012, Laporte et al. 2014, Chierchia et al. 2016). The condition of feature sparsity-involving as few features as possible-requires trading away the data sparsity. This can be achieved by imposing an 1 -norm to the weights w, a classical method to enforce as many zeros in w as can be (Blondel et al. 2013, Combettes & Wajs 2005). In turns, to ensure that the resulting functional can actually be minimized, it is needed to square the hinge loss function F w,b (x, y). Theoretical alternative strategies are investigated in (Chierchia et al. 2016). Further, to account for the severe imbalanced class sizes in the fetal acidosis prediction problem, we propose to modify the penalization term, consisting of a sum N n=1 across all subjects in the database, by further splitting it into two weighted sums according to the class sizes N + and N -, with the additional introduction of an hyperparameter λ. Like for hyper parameter C, its tuning (by cross-validation) is expected to permit to optimize prediction performance.

The functional to minimize to estimate the S-SVM decision rule then reads:

( w, b) ∈ argmin w∈R P , b∈R

w 1 + C λ N - N - n=1 F 2 w,b (x n , y n ) + C (1 -λ) N + N + n=1 F 2 w,b (x n , y n ) (2)
where C controls the tradeoff between feature sparsity and classification performance, and λ ∈ (0, 1) controls the balance between False Positives and False Negatives. In the decision rule

d = sgn(w T x n + b) (3)
only few weights w p are non zero at the price though of increased difficulties to obtain the minimum of (2). Indeed, the non differentiable nature of the 1 -norm precludes the use of gradients and requires that of more involved proximity operators (Blondel et al. 2013, Combettes & Wajs 2005). For fixed hyper-parameters, (C, λ), Eq. 2 is solved in the primal space using a Forward-Backward Splitting Algorithm developed by ourselves.

3.2.3. Hyper-parameter tuning: Cross validation. The selection of hyper-parameters C and λ, as well as the assessment of generalization ‡ performance, are challenging issues. Hyper-parameter (C, λ) selection is here done using single-loop cross-validation (SLCV) [START_REF] Hastie | The Elements of Statistical Learning Springer Series in Statistics[END_REF]). Generalization performance is estimated using double-loop cross-validation (DLCV), nesting and repeating SLCV to obtain realistic and reliable estimates (Spilka et al. 2017, Jonathan et al. 2000).

Following [START_REF] Warrick | [END_REF], Dash et al. 2014), the decision rule threshold b is adjusted to achieve the largest Specificity (SP ) for a Sensitivity (SE) above 0.7, i.e., the smallest number of False Positives (F P ) for at least 70% of True Positives (T P ).

Results: Sparse Learning, labor stages, cross-database evaluation

Using LDB only, Sections 4.1 and 4.2 first describe the benefits of sparse learning and assess differences between S I and S II . Then, Section 4.3 quantifies the ability of the decision obtained from LDB to generalize using the BDB database. 2 reports acidosis prediction performance, for each feature independently and for S I and S II separately, in terms of achieved SP for a targeted SE above 0.70, balanced error rate BER = (SP +SE)/2, TP, FP, and Area under Curve (AUC) computed from the Receiver-Operator-Characteristic (ROC) curves (Fawcett 2006).

For S I , Table 2 shows that features quantifying decelerations achieve the best performance, notably A dec , T stress and MAD dt (FIGO-like) and E VLF (spectral). Interestingly, LTV and STV-classically used to assess FHR variability-yield poor performance, while scale-free features H and c 1 -robust estimates of FHR variability [START_REF] Abry | 26th International Symposium on Computer-Based Medical Systems (CBMS)[END_REF], Doret et al. 2015)-achieve satisfactory performance. This validates both the importance of variability for acidosis prediction and the intuition that variability should not be constructed on specific short or long time-scales, but should rather be based on the scale-free paradigm. Baseline level β 0 also yields satisfactory performance, while baseline trend β 1 surprisingly does not. In addition, spectral features on the LF and HF bands have poor individual power for acidosis predictionconfirming results in (Doret et al. 2015). Moreover, nonlinear scale-free feature c 2 also has satisfactory individual power, while higher-order non linear features {c 3 , c 4 } show much poorer performance.

For S II , Table 2 yields essentially the same conclusions, yet showing that decelerations are better accounted for by T stress -quantifying the percentage of ‡ Performance achieved using data that were never used neither to tune (C, λ) nor to estimate (w, b). underwent contractions-suggesting that too many or too close contractions alter the fetus' well-being. Also, and interestingly, variability is well accounted for by the multifractal feature c 1 , which clearly outperforms the classical LTV and STV FIGOlike features or any spectral feature. In summary, Table 2 shows that single-feature performances are satisfactory for a few FIGO-like features quantifying decelerations, and for scale-free features measuring variability. Yet, none of the features, used as stand-alone decision rule, yield outstanding performance, which motivates the investigation of the performance that can be obtained from joint use of multiple features to construct acidosis prediction rules.

4.1.2. Feature correlations. As a preliminary step, Fig. 2 reports the correlations amongst features. For S I , it shows that features quantifying decelerations-MAD dt , A dec and T stress -are strongly correlated. Further, they are strongly correlated with E V LF , indicating an association of very low frequencies and decelerations. Scalefree features quantifying variability-H, c 1 , h min -are highly correlated, as theoretically expected (H and h min are hence removed from multiple-feature analysis). However, they are only weakly correlated to STV and LTV-thus confirming their different performance. As expected, energies in frequency bands and LF/HF ratio are correlated (Doret et al. 2015). Baseline features do not correlate with any other feature, thus clearly carrying different information. Nonlinear features c 2 , c 3 , c 4 are weakly correlated to linear features.

For S II , observations are essentially identical. Yet, overall correlation is observed to be lower during S II , than during S I , indicating that during S II features measure different aspects of FHR dynamics, while they are mostly related in S I . 4.2. Multiple-feature analysis and Sparse learning 4.2.1. Learning setup. Multiple-feature acidosis detection rules are investigated using the S-SVM methodology described in Section 3. Following preliminary results in (Spilka, Abry, Goncalves & Doret 2014, Spilka et al. 2016b), three decision rules are estimated for S I only, for S II only, and for S I and S II jointly. Hyper-parameter selection was done jointly for (C, λ); their optimal values are shown in Table 3.

Fig. 3 reports, for each setting, the estimated w as a function of the sparsity parameter log 2 C. Hyper-parameter selection was achieved jointly for (C, λ), yet, for ease of exposition, Fig. 3 considers C only. The optimal λ are reported in Table 3 for the sake of completeness, and shown to be close to 0.5 in both cases, suggesting that the mere balance of miss-classifications by class sizes N + /N -is sufficient. Fig. 3 illustrates that the smaller C is, the smaller the number of selected features (i.e., the larger the number of w set to 0).

At the methodological level, Table 3, comparing SLCV and DLCV performances, shows that SLCV performances systematically overestimate those of DLCV, since the same data is used for training and parameter selection.

Stage I.

For S I , Fig. 3 (left) and Table 5 (top) show that optimal performance is achieved involving only a few features: 4 out of the 20 proposed ones. Interestingly, 3 of these are FIGO-like features: MAD dt , T stress and β 0 -the first two associated with decelerations. Further, MAD dt -characterizing deceleration duration and depth-plays a central role. Despite having low performance as a standalone feature, baseline level β 0 is also involved, showing the relevance of this feature, whose nature is different, to complement the decision rule. In addition, variability is also involved in the multiple feature decision rule, yet using the scale-free feature c 1 rather than the LTV and STV ones. Table 5 indicates that acidotic fetuses, compared to healthy ones, are characterized by larger values of MAD dt and T stress (larger impact of decelerations), by a higher baseline β 0 , and a larger c 1 (decrease in variability). Table 3 (line 2) shows that this sparse and optimal decision permits to achieve a SE of 0.66 for a SP of 0.74 (hence BER = 0.70). Table 4 (top panel) reports clinical performance by obstetricians, showing satisfactory Specificity (SP) at the price of a low Sensitivity (SE), and is used as a benchmark. To ease comparisons, Table 4 (mid and bottom panels) also reports S-SVM performance, where the parameter λ has been tuned to match either the SE or the SP of the clinical benchmark, showing the nonnegligible benefit of S-SVM. However, this matched-tuning does not correspond to optimal sparse learning performance, as reported in Table 3. Further comparing Tables 3 and4 clearly indicates that optimal Sparse Learning yields a substantial increase in the tradeoff between SE and SP for both S I and S II . Interestingly, other works based on different feature-selection procedures also concluded that a restricted number of wellselected features were to be preferred to decision rules involving too many features, cf., e.g., (Georgoulas et al. 2017). In summary, classification in S I strikingly selects a sparse decision rule with two (essentially one) features for decelerations, one for baseline, and one for variability-the two first being FIGO-like, the second being scale-free-and hence yielding a decision rule based on the three major pillars in FIGO criteria (baseline, variability, deceleration). Further and complementary comparisons are shared at http://people.ciirc.cvut.cz/ spilkjir/Abry2018BPEXresults.html. 3 to Table 4 indicates that sparse learning improves on the clinical benchmark also for S II . Further, Fig. 3 (middle) shows that optimal performance is obtained for a slightly lower level of sparsity (compared to S I ), hence involving 6 parameters, which further differ from those selected in S I (cf. Table 5 (middle)). Decelerations are still involved, but now T stress and E LF become prominent: the frequency and closeness of decelerations become discriminative rather than their depth. A larger T stress characterizes acidotic newborns, suggesting that too-frequent maternal pushing in Stage II may have negative consequences for the fetus well-being. Also, the number of accelerations #acc is selected and shows a decreased value for acidotic fetuses, confirming that accelerations in FHR remain a sign of good fetal health. Baseline almost does not contribute to the acidosis prediction in S II (only marginally via β 1 ). Variability is also strongly selected, but it is now measured both through c 1 and c 2 : An increased c 1 in acidotic fetuses suggests a lower overall variability, whereas an increased |c 2 | betrays sporadic burstiness and localized transient decreases of variability, within the overall decrease.

C opt C opt C opt |w| log 2 (C) SE ; SP ; AUC log 2 (C) log 2 (C) -9 -8 -7 -6 -5 -9 -8 -7 -6 -5 -9 -8 -7 -6 -5 0.6 0.65 0.7 0.75 c4 c3 c2 c1 α LF/HF E HF E LF E V LF LT V ST V Tstress MAD dt A dec #dec #acc β1 β0

Stage I vs. Stage II. Comparing Table

The fact that different features are selected for S I and S II indicates that temporal dynamics in each stage are different. Though lower than in S I , acidosis prediction performance remains satisfactory (cf. Table 3). Interestingly, Table 3 also shows that the decrease in performance when comparing SLCV and DLCV is larger for StageII, thus suggesting that the learning stage is significantly more difficult, likely caused by a wider inter-individual variety in FHR temporal dynamics for StageII.

For the analysis of S I and S II jointly, Fig. 3 (right) shows that a much larger number of features is required, resembling a combination of those selected independently (cf. Table 5). This confirms that temporal dynamics in S I and S II are actually different, and that a decision rule that tries to be efficient on both stages somehow mixes them up, decreasing overall performance (cf. Table 3).

In summary, since clinicians very well know when the second stage started, there is no reason not to take advantage of that information by using individual, simpler decision rules that are tailored (learned ) for each stage-thus providing a better interpretation.

Sparse Learning and cross-database evaluation

So far, generalization performance, that is performance that would be achieved if the learned decision rule was applied to data that were never used for the training, was evaluated following the classical, yet computationally intensive and practically demanding, DLCV procedure [START_REF] Hastie | The Elements of Statistical Learning Springer Series in Statistics[END_REF], yielding the performance reported in Table 3. Access to two independent databases (LDB and BDB) permits here to evaluate generalization performance in an additional way, by learning the decision rules on the larger LDB and then applying it to the smaller BDB. Table 6 indicates that performance measurements thus achieved are comparable to those using DLCV on the LDB alone. This indicates that the proposed decision rules have a very good generalization ability-a remarkable result since both databases differ in several respects (FHR recording technique, data quality, class imbalance, sampling frequency,. . . cf. Section 2). This suggests that S-SVM yields generalizable decision rules, and that such a robustness comes as a by-product of their sparsity (or simplicity). Table 6 further comforts the relevance of using independent decision rules for both stages: performance using independent rules is much better than for joint ones.

Conclusions and perspectives

The present contribution has quantified the benefits and performance of sparse learning-as implemented by Sparse Support Vector Machines-for fetal acidosis prediction through intrapartum FHR analysis. It has shown that, despite the availability of a large number of features, decision rules only involving a few of them were favored. Interestingly, decision rules essentially involved features associated to the three major groups of FHR characteristics underlying FIGO definitions: Baseline level, decelerations/accelerations, and variability. The present work also shows that FIGOlike features remain competitive compared to those devised from advanced statistical signal processing. However, as pointed out in previous works (Doret, Helgason, Abry, Gonçalvès, Gharib & Gaucherand 2011[START_REF] Abry | 26th International Symposium on Computer-Based Medical Systems (CBMS)[END_REF], Doret et al. 2015)), scale-free parameters H or c 1 (and to a lesser extent c 2 ) provide a far more robust assessment of FHR variability than FIGO's LTV or STV.

Sparse learning also yields decision rules that clearly differ for Stage I and Stage II-evidencing significant differences in their temporal dynamics. Given that the stage of delivery is a naturally available clinical information, there is a significant benefit in using it to design stage-specific decision rules.

Finally, it has been shown that sparse learning achieves very satisfactory generalized performance, in the sense that decision rules learned from one database can be satisfactorily applied to another, even if collected in a different hospital, with different acquisition devices and potentially slightly different clinical practice. This assessment is permitted by the use of several and really different databases, which is rarely reported in the scientific literature.

The satisfactory performances achieved by the proposed sparse learning decision rules, tailored to each stage and validated on an independent database, open the way toward prototype implementations aiming at clinical-practice experimentation.

Future works will also include the assessment of both feature automated computation and acidosis early detection with respect to data quality, notably with respect to the level of missing data in Stage II. This requires further work to devise routines that permits the robust computation of features and robust decision strategies when data quality is poor.

Figure 1 .

 1 Figure 1. Stage splitting. Analyzed FHR data are marked by the time windows framed in rectangles boxes, corresponding to the last 20min of StageI (for StageI dataset, left) and the last 20 min before delivery for StageII dataset (right).

  Acidosis prediction performance. Table

Figure 2 .

 2 Figure 2. Feature correlation. Pairwise correlations for all pairs of features independently for S I (left) and S II (right).

Figure 3 .

 3 Figure 3. S-SVM performance. Feature selection (top row) and classification performance (bottom row) as function of the regularization parameter C, for each of the three sparse learning analyses applied to S I only, S II only and both stages jointly.

Table 1 .

 1 Database and datasets. Clinical data, reported as mean (standard deviation), for the LDB and BDB databases, and S I and S II datasets. Only operative deliveries due to fetal distress only are included.

			LDB		BDB
			Acidotic	Normal	Acidotic	Normal
		# cases	29	1021	14	330
		Umb. cord art. pH	7.01 (0.03)	7.24 (0.07)	7.00 (0.05)	7.26 (0.08)
		Apgar score at 5 min	9.38 (0.90)	9.89 (0.53)	7.71 (1.77)	9.26 (0.89)
	S I	Birth-weight (g) Male/female (n)	3367 (435.31) 3328 (471) 3344 (543.18) 14 (48%) 547 (54%) 10 (71%)	3367 (449) 170 (52%)
		t II (min)	8.5 (5.2)	6.8 (5.1)	10.3 (4.8)	9.3 (3.9)
		# Operative delivery	13 (45%)	213 (21%)	-	-
		# cases	27	735	12	116
		Umb. cord art. pH	7.01 (0.04)	7.22 (0.06)	6.99 (0.05)	7.22 (0.07)
		Apgar score at 5 min	9.56 (0.80)	9.90 (0.43)	7.67 (1.44)	8.91 (1.04)
	S II	Birth-weight (g) Male/female (n)	3469 (397) 13 (48%)	3365 (444) 392 (53%)	3265 (437) 5 (42%)	3421.81 (405) 61 (53%)
		# t II ≤ 15 min (n)	27.9 (9.8)	27.6 (9.8)	25.0 (4.3)	23.7 (4.3)
		Operative delivery(n)	13 (48%)	152 (21%)	-	-
	umbilical artery and venous pH after delivery and decision to intervene due to suspected
	fetal acidosis, cf. (Doret, Massoud, Constans & Gaucherand 2011) for details.
	For BDB, 552 FHR recordings were acquired at the Hospital in Brno, from 2010
	to 2012, using STAN S21 or S31 scalp electrodes, or Avalon FM40 or FM50 Doppler-
	based devices (Phillips Healthcare, Andover, MA), via either ultrasound probes or scalp
	electrodes (12 bit resolution, sampling rate 4 Hz). BDB has been made an open-access
	database

Table 2 .

 2 Fetal acidosis prediction: univariate performance

			S I				S II
	Feature: AUC SE SP BER TP FP AUC SE SP BER TP FP
	β 0	.65	.72 .53	.63	21 477	.51	.70 .18	.44	19 600
	β 1	.54	.72 .26	.49	21 760	.55	.70 .29	.50	19 523
	#acc	.50	.72 .17	.45	21 844	.55	.70 .30	.50	19 513
	#dec	.60	.72 .41	.56	21 608	.50	.70 .29	.50	19 518
	A dec	.71	.72 .59	.66	21 420	.56	.70 .36	.53	19 468
	MAD dt	.76	.72 .64	.68	21 363	.63	.70 .49	.60	19 371
	T stress	.73	.72 .69	.71	21 315	.71	.70 .58	.64	19 306
	LTV	.53	.72 .28	.50	21 737	.52	.70 .22	.46	19 570
	STV	.51	.72 .25	.49	21 767	.51	.70 .30	.50	19 517
	LF/HF	.59	.72 .49	.61	21 516	.50	.70 .30	.50	19 517
	E VLF	.75	.72 .68	.70	21 329	.62	.70 .51	.60	19 363
	E LF	.60	.72 .38	.55	21 633	.55	.70 .46	.58	19 394
	E HF	.50	.72 .28	.50	21 738	.58	.70 .47	.59	19 389
	α	.58	.72 .45	.59	21 563	.51	.70 .28	.49	19 530
	h min	.67	.72 .53	.63	21 481	.62	.70 .47	.59	19 390
	c 1	.69	.72 .47	.60	21 541	.71	.70 .68	.69	19 236
	c 2	.66	.72 .54	.63	21 475	.66	.70 .50	.60	19 366
	c 3	.53	.72 .28	.50	21 731	.50	.70 .22	.46	19 573
	c 4	.50	.72 .10	.41	21 916	.64	.70 .51	.61	19 358
	H	.69	.72 .60	.66	22 403	.67	.70 .60	.65	19 295

Table 3 .

 3 Acidosis prediction performance -LDB Database. Performance are reported for DLCV (top) and SLCV (bottom) and different pairs of training / testing datasets.

	Tr / Te log 2 C λ	SE	SP	BER #TP #FN #TN #FP
	S I /S I	-8	.49	.66	.74	.70	19	10	751 270
	S II /S II -6.5	.51	.59	.71	.65	16	11	520 215
	S/S	-6	.47	.71	.66	.69	40	16	1160 596
	S I /S I	-8	.48	.71	.73	.72	21	8	743 278
	S II /S II -7	.48	.72	.70	.71	20	7	514 221
	S/S	-6.5	.47	.72	.67	.70	41	15	1179 577

Table 4 .

 4 Fetal acidosis prediction. Performance for S I and S II , computed from clinical information (top panel, used as benchmark), and compared to those obtained from S-SVM classifier tuned to match the either the SE or the SP of the clinical benchmark (middle and bottom panels, respectively).

			SE SP BER #TP #FP #FN #TN
	Clinical	S I	.45 .79	.62	13	213	16	808
	benchmark	S II	.48 .79	.64	13	152	14	583
	S-SVM,	S I	.45 .89	.67	13	16	912	109
	matched SE	S II	.48 .86	.67	13	14	632	103
	S-SVM,	S I	.62 .79	.71	18	11	811	210
	matched SP	S II	.56 .79	.68	15	12	586	149

Table 5 .

 5 Selected feature statistics. Median (maximum absolute deviation) per class for each selected feature in S I only, S II only, and both decision rules (S).

		name	w	Acidotic	Normal
		MAD dt	.90 20.3 (7.1)	10.1 (5.0)
	S I	β 0 c 1	.38 .18 0.65 (0.14) 0.54 (0.11) 156 (10) 147 (12)
		T stress	.08 0.63 (0.13) 0.43 (0.20)
		T stress	.63 0.59 (0.09) 0.53 (0.13)
		c 1	.62 0.79 (0.12) 0.64 (0.13)
	S II	|c 2 |	.32 0.19 (0.09) 0.16 (0.07)
		E LF	.30 36.2 (16.1) 39.7 (21.7)
		#acc	.14	0.0 (0.6)	0.0 (0.9)
		β 1 (•1E5) .05	5.8 (94)	-7.5 (94)
		T stress	.74 0.61 (0.11) 0.49 (0.18)
		c 1	.36 0.71 (0.14) 0.58 (0.12)
	S	|c 2 |	.30 0.15 (0.08) 0.10 (0.07)
		MAD dt	.27 20.4 (6.7)	13.6 (5.8)
		E HF	.26 5.14 (3.48) 5.20 (4.29)
		β 0	.19	153 (13)	147 (12)
		E LF	.17 25.2 (15.9) 22.5 (19.7)
		β 1 (•1E5) .15	9.9 (74)	-2.1 (71)

Table 6 .

 6 Generalization performance. Learning on the LDB Database, performance evaluation on the BDB database.

	Train/Eval AUC SE SP BER TP FN TN FP
	S I /S I	.78	.64 .80	.72	9	5 263 67
	S II /S II	.73	.58 .77	.68	7	5	89 27
	S/S	.72	.42 .81	.62	11 15 361 85
	S/S I	.66	.21 .86	.54	3	11 283 47
	S/S II	.76	.67 .67	.67	8	4	78 38

Acknowledgments

Work supported by Grant ANR-16-CE33-0020 MultiFracs.

Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum provided under the programme Projects of Large Research, Development, and Innovations Infrastructures (CESNET LM2015042) is greatly appreciated.

Appendix A. Abbreviation and Symbol list


Emmanuel Manu Dabankah
email: emmanuel-manu.dabankah@insa-lyon.fr

Nicolas Totaro

Jérôme Antoni

Blind identification using inverse Patch Transfer Function (iPTF) method

Keywords: acoustic field reconstruction, blind identification, inverse Patch Transfer Function

An important object of vibration and noise reduction and control is to identify and localize the vibration and noise sources. Many inverse methods, like Nearfield Acoustic Holography, have been developed in acoustics in the last decades. Among others, the iPTF method, allows the reconstruction of the acoustic fields (pressure, velocity, intensity) directly on the vibrating structure surface even when it has a complex shape. In addition, measurements can be done in non-controlled acoustic environments. The concept of iPTF is based on the application of Green's identity on any closed virtual volume defined around the source. The reconstruction of sound source fields combines discrete acoustic measurements performed at accessible positions around the source with acoustic impedance matrices. In the present work, blind identification of the vibratory fields is proposed. The "blindness" has here two meanings: -the identification of the velocity field of a vibrating structure can be blind if obstacles mask parts of the structure to characterize.

-the identification can be blind if the velocity field is the result of the combination of several unknown sources and if one wants to separate the contribution of each source. Some numerical and experimental results will be shown to illustrate both aspects of the blind identification.

INTRODUCTION

The source identification in the presence of obstacles is a matter of relevance for industrial applications. In automotive industry, it would be valuable to retrieve t he velocity field of a source that is intricately located with other structures partially blocking its accessibility. This situation is mostly encountered in the gearbox and engine compartments of a vehicle. In the present study, a method called "inverse Patch Transfer Function" (iPTF) (1-3) is used to handle this task.

The iPTF method is able to identify the vibration velocity of irregularly shaped sources in a noncontrolled acoustic environment (reverberant room, stationary disturbing sources, etc.). The objective of this present work is to demonstrate numerically how to use the iPTF method to identify the velocity field of a source masked by a rigid obstacle. A test case is proposed where an obstacle located at 1 cm in front of the vibrating structure is masking 45% of its surface. The remarkable quality of the reconstructed field compared to the reference computation demonstrates the capability of iPTF method to handle the presence of masking obstacles.

In addition, introduction of blind separation techniques in iPTF method should allow to distinguish velocity fields produced by different vibration sources. The last section is dedicated to show how the approach is used to separate the various phenomena that lead to the noise generation.

THEORETICAL BACKGROUND OF THE IPTF METHOD

The following example in Figure 1 gives a brief description of iPTF method. Details of theoretical derivation have been presented in Ref. [START_REF] Forget | Source fields reconstruction with 3D mapping by means of the virtual acoustic volume concept[END_REF]. Let assume the source represented by the surface vibrates with a known velocity field in a semiinfinite medium. The iPTF method is based on the definition of an arbitrary virtual acoustic cavity surrounding the source to identify (see Figure 1). The virtual acoustic cavity Ω engulfs an obstacle of surface '''. This virtual cavity is thus delimited by the surfaces ', '' and ''' with corresponding points Q', Q'' and Q''' (see Figure 1). Moreover, disturbing (stationary) sources might exist but must be excluded from the virtual cavity Ω. The acoustic problem is governed, in this acoustic domain, by the Helmholtz equation. On boundaries, a normal harmonic particle velocity is imposed on the source surface, while the normal harmonic particle velocity on the virtual acoustic surface results from the radiation of the source in the acoustic medium. The corresponding particle velocities on the obstacle surface and platform surface are zero. This is because they are assumed to be rigid. Consequently, the virtual cavity problem to solve is given by Equation 1.

(1) where is the Laplacian operator, the normal derivative outwardly directed, is the angular frequency, is the acoustic pressure and is the acoustic wavenumber. As detailed in Refs. (1-3), this system of equations can be solved introducing acoustic impedance transfer functions and dividing the surfaces into element surfaces called patches. Therefore, the pressure at point can be expressed as [START_REF] Totaro | Sound fields separation and reconstruction of irregularly shaped sources[END_REF] where [START_REF] Forget | Source fields reconstruction with 3D mapping by means of the virtual acoustic volume concept[END_REF] The subscripts , and denote respectively a point inside the virtual volume, a patch of the source surface Σ and a patch of the virtual surface Σ′′.

are the mode shapes of the virtual cavity with rigid walls.

is the norm of the n-th mode shape. The notation represents the space average of the variable on patch and is the surface of the patch . is the real acoustic wavenumber at the eigen angular frequency , is the complex wave number. The terms and are the acoustic impedances. Equation (2) permits the calculation of the pressure at a point using the velocities of the patches. For a sake of simplicity, Equation 2 is rewritten under matrix form when handling several points i in the virtual cavity (4) The aim of the iPTF method is to identify the source velocities . For that purpose, one uses Equation 4, which after simple matrix manipulation allows to compute source velocities as (5) The acoustic patch impedance matrices of the volume defined by surfaces Σ and Σ′′ are computed using Equation 3 and the eigenmodes extracted with a standard finite element solver. The pressure and velocity vectors of Equation 5are measured in the virtual acoustic cavity and on the virtual surface Σ′′ respectively. The matrix is often ill-conditioned and a regularization technique has to be used to invert it. The solution is found using Tikhonov regularization and the maximum of curvature of the L-curve is used to define the best regularization parameter -e.g. see Refs. [START_REF] Totaro | Sound fields separation and reconstruction of irregularly shaped sources[END_REF][START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems[END_REF][START_REF] Leclère | Acoustic Imaging Using Under-Determined Inverse Approaches: Frequency Limitations and Optimal Regularization[END_REF].

NUMERICAL VALIDATION

For this numerical validation, the identification is performed on a simply supported plate excited by a harmonic point force as presented in Figure 2. The plate is 0.6m long, 0.4m wide, and 2mm thick, and it is made of aluminium (Young's modulus Pa; density 2700 kg/m 3 ; Poisson's ratio 0.35; damping 0.01). The plate is excited by a unit point force located at point (0.15; 0.15) m on the frequency band [100:500] Hz with frequency step of 25 Hz.

In the following example, the plate is partially masked by a rigid obstacle. As can be seen in Figure 2, this obstacle has an ovoid shape and is located really close to the plate (1 cm). This impl ies that no measurement can be done in this non-accessible zone. The aim of this example is to demonstrate that the iPTF method is able to handle the presence of an obstacle and still able to reconstruct the velocity field on the surface of the masked structure.

Measurement Step (Numerical Experiment)

To identify the source velocity field using the iPTF method, it requires measurement and numerical data. The measurement data are the set of pressure measurement s taken randomly inside the virtual volume and the set of velocity measurements on the virtual surface patches. These measurements are usually done by physical experiments, but in present work they are simulated. The radiated field obtained at the measurement step has been provided by a numerical experiment performed using ACTRAN software (6).

The virtual acoustic surface Σ'' is made of the surfaces of a rectangular box (0.8 x 0.6 x 0.4) m open to the bottom. It is divided into 416 patches of size 0.05m. The plate is divided into 600 patches each of size 0.02 m. A set of 600 random field points are selected such that they are a least 0.02 m from the surfaces of the source and virtual cavity. In all there were 416 velocity measurement points (in red) on the virtual surface and 600 pressure measurement points (in blue) in the acoustic cavity. The number of measurement points is voluntarily high to avoid questions raised by the use of an underdetermined problem. The obstacle is positioned between the source and these measurement points as shown in Figure 2. The pressure and velocity quantities are recorded from their corresponding field points as shown in Figures 3 and4 respectively. The second set of data are the numerical data which are obtained by performing modal response analysis and finite element analysis of the virtual volume and its associated boundary surfaces (virtual surfaces and source surface) and the chosen random field points. The Neumann boundary condition is imposed on all boundary surfaces (2). This implies that the surrounding surfaces of the virtual cavity are assumed rigid. Obviously, this virtual cavity with rigid wall does not physically exist, it is only chosen for theoretical convenience as a basis for acoustic impedance decomposition (3). The eigenmodes of the virtual cavity with these rigid walls are computed up to 4250Hz (971 modes).

Identification Step

The velocity field is identified using Equation ( 5) and the previously detailed data. It is important to note that the surface of identification corresponds to the surface of the plate, thus from the principle of continuity, the particle velocity is equal to the structural velocity of the plate. The computed plate and identified velocity fields are compared in terms of the norm velocity as presented in Figure 8. These results demonstrate that even in such an extreme case (45% of the structure is masked by a very close rigid object), the iPTF process can provide satisfactory results for this kind of blind identification.

SOURCE SEPARATION TECHNIQUES

In general, the iPTF method is able to identify radiated acoustic field based on phenomena leading to its generation. It is of interest to separate the partial contributions of the different phenomena that contribute to the overall radiated acoustic field. The phenomena in this context relates to the different sound sources (the mechanisms that lead to the excitation). A practical industrial example is the engine noise in a car. This radiated engine noise results from the mixt ure of many complex noises produced by various sources such as combustion, injection, compression and exhaustion. These are referred to as the phenomena. Pertaining to this work, the different phenomena will be blindly separated (which implies that there is no prior information about these phenomena). Some approaches, known as Blind Source Separation (BSS) [START_REF] Jutten | Blind separation of sources, Part I: An adaptive algorithm based on neuromimetic architecture[END_REF][START_REF] Kokkinakis | Advances in Modern Blind Signal Separation Algorithms: Theory and Applications[END_REF] techniques will be evaluated in the context of iPTF method. BSS involves the task of 'blindly' recovering a set of unknown signals, the so-called sources from their observed mixtures, based on very little to almost no prior knowledge about the source characteristics or the mixing structure [START_REF] Comon | Independent component analysis: A new concept[END_REF]. The goal of BSS is to process multi-sensory observations of an inaccessible set of signals in a manner that reveals their individual (and original) form, by exploiting the spatial and temporal diversity, readily accessible through a multi-microphone configuration [START_REF] Kokkinakis | Advances in Modern Blind Signal Separation Algorithms: Theory and Applications[END_REF]. Algebraically, most linear BSS models can be expressed as the following specific matrix factorization [START_REF] Cichocki | Blind signal processing methods for analyzing multichannel brain signals[END_REF] Y = AX +W (6) where Y is m×n observed mixture matrix, A is m×r mixing matrix, X is r×n source signal matrix, W is m×n noise signal matrix, and m, r and n stand for the number of observations, the number of sources, and the number of samples, respectively.

In present work, two ways to implement BSS are proposed. The first approach involves measuring acoustic field signals (particle velocities and pressure) with set of microphones directly from a vibrating structure. The measured acoustic data is used as the input Y, the observed mixture matrix. With the use of appropriate BSS method, X, the source signal matrix is retrieved. The second approach is similar to the CLEAN algorithm [START_REF] Hogbom | Aperture synthesis with a non-regular distribution of interferometer baselines[END_REF][START_REF] Hutchinson | Investigation of radio astronomy image processing techniques for use in the passive millimetre-wave security screening environment[END_REF] and an overview to this approach is illustrated in Figure 9. The proposed algorithm aims to extract the sources using the iPTF velocity map and source prior information (sources coupling level and location). The iPTF velocity map can be viewed as a collection of patched sources. The quest is to recover the source image from the iPTF velocity map using a set of imaging processing techniques. The present sources are assumed to be spatially disjointed. The process begins with the identification of the patch with the highest energy level in the iPTF map. In an iterative process, this highest energy patch is subtracted from the iPTF velocity map. A delta function whose peak intensity and location equals highest energy patch (known as a source component) is saved to a separate image. What remains of the iPTF velocity map distribution (the 'residual' image) is fed back into the iterative loop and the cycle repeats until a stopping criterion is reached; usually a predetermined threshold value. By this method, the source components build up as a series of delta functions whose positions and intensities directly relate to the patched sources of iPTF velocity map. The source components and the residual image are combined using the k-means clustering algorithm [START_REF] Lloyd | Least squares quantization in PCM[END_REF] to generate the final source image.

CONCLUSIONS

In this article, iPTF has been used to identify velocity fields in the presence of obstacle. The main advantage of the method is the combined use of the integral formulation, FEM and measurements, which allows to overcome inherent limitations of classical methods. This paper gave an overview of the theoretical background and the numerical measurement methodology. To prove the method efficiency, a numerical validation was set up with a simple supported plate partially masked by a rigid obstacle. This identification has been compared to plate structural velocity. The results show a good comparison, which confirms the expected advantages, and demonstrate the applicability of the iPTF method.

The last part of the presented article presents a first thought on how to introduce blind source separation techniques in the context of iPTF method.

Figure 1 -

 1 Figure 1 -Normal velocity reconstruction of a plate in a noisy environment in the presence of an obstacle

Figure 2 -Figure 3 -Figure 4 -

 234 Figure 2 -Geometric set up in ACTRAN software used in the acoustic radiation generation (blue points: field points in the virtual volume; red points: field points on the centroid position of the virtual surface patches; orange ovoid: obstacle; grey rectangle: plate; red arrow: point force; blue triangle: plate support; grid box: virtual surface patches). (a) 3D view; (b) Y-Z view.

Figure 5 -Figure 7 -

 57 Definition of the virtual acoustic cavity model. (a) unmeshed model of the virtual acoustic cavity; (b) 3D volume mesh of virtual acoustic cavity; (c) cut through of the virtual acoustic volume mesh revealing the position of the obstacle. Definition of the surfaces delimiting the virtual acoustic cavity used in the modal response analysis. (a) upper part of the 2D meshed virtual surface; (b) lower part of the 2D meshed virtual surface. (c) 2D surface mesh of the source. The real part of the modal pressure maps at an eigenfrequency of 353Hz. (a) pressure map of the upper part of the 2D meshed virtual surface; (b) pressure map of 2D meshed source surface mesh; (c) the random field points inside the virtual volume.

Figure 8 -

 8 The norm of the velocity field (a) computed on plate (reference) at 100Hz; (b) identified by iPTF at 100Hz; (c) computed on plate (reference) at 175Hz; (d) identified by iPTF at 175Hz; (e) computed on plate (reference) at 475Hz; (f) identified by iPTF at 475Hz. The tilted ellipse is the outline projection of the obstacle onto the plate.

Figure 9 -

 9 Figure 9 -Flowchart of the proposed separation algorithm routine with illustrative images. This is an image processing technique of the iPTF map.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the European Commission for its support of the Marie Sklodowska Curie program through the ETN PBNv2 project (GA 721615).

This work was performed within the framework of the LABEX CeLyA (ANR-10-LABX-0060) of Université de Lyon, within the program « Investissementsd'Avenir » (ANR-16-IDEX-0005) operated by the French National Research Agency (ANR).