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Due to its possible low-power implementation, Compressed Sensing (CS) is an attractive tool for physiological signal acquisition in emerging scenarios like Wireless Body Sensor Networks (WBSN) and telemonitoring applications. In this work we consider the continuous monitoring and analysis of the fetal ECG signal (fECG). We propose a modification of the low-complexity CS reconstruction SL0 algorithm, improving its robustness in the presence of noisy original signals and possibly ill-conditioned sensing/reconstruction procedures. We show that, while maintaining the same computational cost of the original algorithm, the proposed modification significantly improves the reconstruction quality, both for synthetic and real-world ECG signals. We also show that the proposed algorithm allows robust heart beat classification when sparse matrices, implementable with very low computational complexity, are used for compressed sensing of the ECG signal.

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- 1. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it CompressionandBeyond ROBUSTRECONSTRUCTIONFORCS-BASEDFETAL BEATSDETECTION Giulia Da Poian dapoian.giulia@spes.uniud.it Riccardo Bernardini bernardini@uniud.it Roberto Rinaldo rinaldo@uniud.it University of Udine Polytechnic Department of Engineering and Architecture Via delle Scienze 206, Udine, Italy See also: http://ieeexplore.ieee.org/document/7305770/ http://www.mdpi.com/1424-8220/17/1/9/htm
- 2. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it ROBUSTRECONSTRUCTIONFORCS-BASEDFETALBEATSDETECTION • We propose a novel system for the compression and analysis of Abdominal Fetal Electrocardiogram using Compressive Sensing (CS) and Independent Component Analysis (ICA) applied in the compressed domain, and sparse representations in a specific dictionary. • We describe the proposed scheme and a robust variant of the Smoothed-L0 reconstruction algorithm. • The proposed modification significantly improves the reconstruction quality, both for synthetic and real-world ECG signals. Roberto Rinaldo August 31st, 2016
- 3. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Roberto Rinaldo August 31st, 2016
- 4. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Outline • Overview of Sparse Representations and Compressive Sensing (CS) • Gaussian Dictionary for ECG approximation and CS applied to ECG signal • Analysis of non invasive Fetal Electrocardiogram (fECG) - adopted methodologies • Reconstruction algorithm • Results Roberto Rinaldo August 31st, 2016
- 5. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Sparse Representation coefficients basis, frame Many (blue) Sparse representation of an image via a multiscale wavelet transform Approximation of image obtained by keeping only the largest 10% of the wavelet coefficients. A signal is sparse if most of its coefficients are (approximately) zero Signals can often be well-approximated as a linear combination of just a few elements from a known basis or dictionary Roberto Rinaldo August 31st, 2016
- 6. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it The column vectors of a dictionary are discrete time elementary signals called dictionary atoms Sparse Representation Taking Advantage of Sparsity for: audio/image/video signal detection and classification blind source separation • compression • denoising • superresolution To improve sparsity of composite signals, one has to construct a transform matrix with the best basis DICTIONARY: collection of elementary waveforms or atoms or basis functions. Example of an Overcomplete Dictionary 250 x 4267) Roberto Rinaldo August 31st, 2016
- 7. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it COMPRESSIVE SENSING (CS): signals that are sparse in some domain, can be fully reconstructed using only few random measurements. • Asymmetrical: Most processing at decoder • Universality: Random measurements can be used for signals sparse in any basis Compressive Sensing Systems adopting compressive sensing can: achieve sub-Nyquist sampling rates directly acquire compressed representations of signals process signals and solve inference problems in a reduced-dimensionality domain with small or no penalties Random Sensing Matrix Measurements Roberto Rinaldo August 31st, 2016
- 8. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it CompressiveSensing-Reconstruction Since M<N there are infinitely many solutions How to solve the undetermined system of equation to recover the original signal from the measurements vector? Signal reconstruction algorithm aims to find signal’s sparse coefficient vector NP- hard l0 norm minimization: can reconstruct the signal exactly with high probability using only M=k+1 measurements l1 norm minimization: can exactly recover k-sparse signals using M=c * k log(N/K) measurements Roberto Rinaldo August 31st, 2016
- 9. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Outline • Overview of Sparse Representations and Compressive Sensing (CS) • Gaussian Dictionary for ECG approximation and CS applied to ECG signal • Analysis of non invasive Fetal Electrocardiogram (fECG) - adopted methodologies • Reconstruction algorithm • Results Roberto Rinaldo August 31st, 2016
- 10. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it GaussianDictionary Traditional approaches use analytical sparsifying transform (e.g. DWT, DCT, …) to sparse represent the ECG signals Limited Compression Ratios Limited Reconstruction Quality Exploit the sparsity of the ECG signal Study the morphology of the PQRST cycle Scale (Shape) parameter Shift parameter Symmetric waves (Q, R and S) can be approximated by one Gaussian function Asymmetric waves (P or T) require 2-3 functions Roberto Rinaldo August 31st, 2016
- 11. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it GaussianDictionary Dictionary atom Coefficient Real ECG reconstruction examples from M=62 measurements (CR=75%) using a sparse binary sensing matrix Using Gaussian Dictionary Reconstruction quality PRD=5.48% Using Wavelets Reconstruction quality PRD= 28.19% Roberto Rinaldo August 31st, 2016 the RR measure metric is used, which is calculated from the differences between matched reference RR and test RRd intervals. This metric is denoted here by RRmeas (ms) RRmeas = v u u t 1 I 1 I 1X i=1 (RRi RRd i )2, (18) where I is the total number of fetal QRS complexes in the reference. These measures correspond to the Physionet Challenge scores and were obtained with the same code used by the Challenge scorer. Since we are applying a compression technique (CS), recon- struction quality is evaluated using the PRD metric, deﬁned as PRD(%) = sP n(x(n) ˆx(n))2 P n x(n)2 ⇥ 100, (19) where, x(n) and ˆx(n) are the original (after baseline wander removal and notch ﬁltering) and reconstructed signals, respec- tively. This value is computed for each reconstructed segment of every channel and then the average value is calculated. According to [38], reconstructions with PRD values between 0% and 2% are qualiﬁed to have “very good” quality, while values between 2% and 9% are categorized as “good”. Finally, we consider the total time required by the algorithm for beat classiﬁcation, including reconstruction of all the 4 channels, in order to asses the possibility to implement the proposed framework in a real-time application. The average time required by the algorithm, for a 1 minute long signal, is approximately 3.7 s. The reconstruction program is written in Matlab, running on an Intel Core i7 processor, equipped with 16 GB memory. RESUL FROM THE S Record r01 r04 r07 r08 r10 Conce Challe A, i Challe record have a the di a min and a predic value positiv for the respec To the ab differe compr results percentage root-mean-square difference
- 12. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it GaussianDictionary Proposed Gaussian Dictionary is independent from the training set dose not require any pre-processing Increases the compression of: 25% with respect to CS with DWT 7% with respect to BSBL-BO Roberto Rinaldo August 31st, 2016 Bounded-block-Optimized Block Sparse Bayesian Learning (BSBL-BO) Orthogonal Matching Pursuit (OMP) Basis Pursuit Denoising (BPDN)
- 13. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Outline • Overview of Sparse Representations and Compressive Sensing (CS) • Gaussian Dictionary for ECG approximation and CS applied to ECG signal • Analysis of non invasive Fetal Electrocardiogram (fECG) - adopted methodologies • Reconstruction algorithm • Results Roberto Rinaldo August 31st, 2016
- 14. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Abdominal fetalECG Heart defects are among the most common birth defects and leading cause of birth defect-related deaths Noninvasive FECG monitoring makes use of electrodes placed on the mother's abdomen Recorded signals are a mixture of Maternal ECG, Fetal ECG and noise (Respiration, EMG …) Fetal QRSMaternal QRS Roberto Rinaldo August 31st, 2016
- 15. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it CSofabdominalfetalECG Fetal ECG recorded on the abdomen has a low SNR 5-1000 times smaller in intensity than in the adult Less sparse than adult ECG, for example in the Wavelet domain Reconstruction does not have to affect the interdependence relation among the multichannel recordings Current CS algorithms generally fail in this application Gaussian Dictionary can be used to increase the performance of CS applied to fetal ECG Roberto Rinaldo August 31st, 2016
- 16. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it ICAintheCSdomain Independent Component Analysis ICA for the separation of mixed signals (source signals are independent and have non-gaussian distributions) We propose to perform ICA directly on the compressed measurements Compressed Independent Components 4 x m Original mixture signal (4 channels) Reconstructed ICs from ICA applied in CS domain ICs from ICA applied on original mixture Abdominal Signals 4 x N Independent Components 4 x N Mixing Matrix 4 x 4 Roberto Rinaldo August 31st, 2016
- 17. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it BeatsClassification Information about time and frequency location of maternal and fetal QRS complex The first part of the dictionary is for Maternal ECG approximation The second part of the dictionary for Fetal ECG approximation Classification is based on the atoms activated during reconstruction Decomposition of IC signal in the dictionary part related to maternal approximation Decomposition of IC signal in the dictionary part related to fetal approximation Roberto Rinaldo August 31st, 2016
- 18. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it BeatsClassification Detected Maternal QRS of the first ICA signal Detected Fetal QRS of the second ICA signal One more example of the proposed detection method: in red fetal beats and in green maternal beats (on one of the 4 original signals) Roberto Rinaldo August 31st, 2016
- 19. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Outline • Overview of Sparse Representations and Compressive Sensing (CS) • Gaussian Dictionary for ECG approximation and CS applied to ECG signal • Analysis of non invasive Fetal Electrocardiogram (fECG) - adopted methodologies • Reconstruction algorithm • Results Roberto Rinaldo August 31st, 2016
- 20. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Reconstructionalgorithm Roberto Rinaldo August 31st, 2016 • Reconstruction of the independent components is done using a modiﬁed version of the SL0 reconstruction algorithm, introducing regularization for better immunity against noise (D has D atoms) random variables [2], sparse binary matrices, where has only d non-zero randomly selected entries in each column, have been proposed to reduce the computational cost [8]. In this case, calculating x takes only O(dN) operations, with a signiﬁcant saving when d ⌧ N. The SL0 algorithm proposed in [6] solves the problem in Eq. (1) by approximating the l0-norm with a continuous func- tion, and optimizing the resulting cost function to provide a smooth measure of sparsity. Indeed, the l0-norm can be ap- proximated using Gaussian functions, for small values [6], as in ||s||S,0 , D DX i=1 exp( s2 i /2 2 ). (2) Thus, the minimization of the l0-norm is approximately equivalent to maximize F (s) = P i exp( s2 i /2 2 ). This enables to replace the l0-norm minimization with a convex problem, and maximize F (s) using a steepest ascent algo- rithm. The parameter controls the trade-off between the smoothness of the objective function and the accuracy of the approximation of the l0-norm. The algorithm proposed in [6] consists of two nested it- erations, and the external loop is responsible to gradually de- crease the value. Note that, when is sufﬁciently large, exp( s2 /2 2 ) ⇡ 1 s2 /2 2 , and the maximization of F (s) ill-conditioned, th and results in po SL0 proposed in [ optimization prob As in the SL0 using (2), and the erations. The inte feasible set {s| k ˜s = s µ k and p min ˆs k ˆs Using the Lagrang rewritten as min ˆs where is the reg ˆs = ˜s As for the SL0 is equal to the l2 Solving the proble • SL0: approximate the L0 norm with the smooth function • Problem (in the noisy case: min s ||s||0 s.t. As = y, A = D F (s) = DX i=1 exp( s2 i /2 2 ) • Iterate decreasing ! to approach the L0 norm, and use a gradient based steepest ascend procedure to maximize • At each iteration, project back to the feasible set via s s AT (AAT ) 1 (As y) ||y As||2 ✏ )
- 21. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it • SL0 requires that matrix AAT is invertible, and this can be problematic with large M (low compression ratio) Reconstructionalgorithm Roberto Rinaldo August 31st, 2016 • λSL0: approximate the L0 norm with the SL0 smooth function • Iterate decreasing ! to approach the L0 norm, and use a gradient based steepest ascend procedure to maximize • At each iteration, project back to the feasible set F (s) es, where has in each column, onal cost [8]. In operations, with es the problem in continuous func- tion to provide a norm can be ap- mall values [6], 2 ). (2) is approximately s2 i /2 2 ). This n with a convex pest ascent algo- -off between the e accuracy of the of two nested it- ill-conditioned, then application of A† ampliﬁes the error and results in poor reconstruction, even using the Robust SL0 proposed in [9]. Introducing a regularization term in the optimization problem enables a stable recovery of x = Ds. As in the SL0 algorithm, we approximate the l0-norm by using (2), and the algorithm again consists in two nested it- erations. The internal loop seeks the maximum of F in the feasible set {s| k y As k2 ✏}. At each step we compute ˜s = s µ k and project ˜s by solving min ˆs k ˆs ˜s k2 s.t. k Aˆs y k2 ✏. (5) Using the Lagrangian function of Eq. (5), the problem can be rewritten as min ˆs k Aˆs y k2 2 + k ˆs ˜s k2 2, (6) where is the regularization parameter. The solution is ˆs = ˜s AT (AAT + IM ) 1 (A˜s y). (7) As for the SL0 algorithm, for large values, the solution ces, where has s in each column, ational cost [8]. In ) operations, with ves the problem in a continuous func- ction to provide a 0-norm can be ap- small values [6], 2 2 ). (2) m is approximately ( s2 i /2 2 ). This ion with a convex epest ascent algo- e-off between the he accuracy of the s of two nested it- le to gradually de- sufﬁciently large, imization of F (s) ill-conditioned, then application of A ampliﬁes the error and results in poor reconstruction, even using the Robust SL0 proposed in [9]. Introducing a regularization term in the optimization problem enables a stable recovery of x = Ds. As in the SL0 algorithm, we approximate the l0-norm by using (2), and the algorithm again consists in two nested it- erations. The internal loop seeks the maximum of F in the feasible set {s| k y As k2 ✏}. At each step we compute ˜s = s µ k and project ˜s by solving min ˆs k ˆs ˜s k2 s.t. k Aˆs y k2 ✏. (5) Using the Lagrangian function of Eq. (5), the problem can be rewritten as min ˆs k Aˆs y k2 2 + k ˆs ˜s k2 2, (6) where is the regularization parameter. The solution is ˆs = ˜s AT (AAT + IM ) 1 (A˜s y). (7) As for the SL0 algorithm, for large values, the solution is equal to the l2 norm solution subject to k y As k2 ✏. Solving the problem 2 2 . Let be the updated solution. s, where has n each column, onal cost [8]. In operations, with s the problem in ontinuous func- ion to provide a norm can be ap- all values [6], 2 ). (2) s approximately s2 i /2 2 ). This n with a convex est ascent algo- off between the accuracy of the f two nested it- to gradually de- ufﬁciently large, ill-conditioned, then application of A† ampliﬁes the error and results in poor reconstruction, even using the Robust SL0 proposed in [9]. Introducing a regularization term in the optimization problem enables a stable recovery of x = Ds. As in the SL0 algorithm, we approximate the l0-norm by using (2), and the algorithm again consists in two nested it- erations. The internal loop seeks the maximum of F in the feasible set {s| k y As k2 ✏}. At each step we compute ˜s = s µ k and project ˜s by solving min ˆs k ˆs ˜s k2 s.t. k Aˆs y k2 ✏. (5) Using the Lagrangian function of Eq. (5), the problem can be rewritten as min ˆs k Aˆs y k2 2 + k ˆs ˜s k2 2, (6) where is the regularization parameter. The solution is ˆs = ˜s AT (AAT + IM ) 1 (A˜s y). (7) As for the SL0 algorithm, for large values, the solution is equal to the l2 norm solution subject to k y As k2 ✏. Solving the problem • Equivalently, solve atrices, where has ries in each column, utational cost [8]. In dN) operations, with solves the problem in ith a continuous func- function to provide a e l0-norm can be ap- or small values [6], s2 i /2 2 ). (2) orm is approximately exp( s2 i /2 2 ). This zation with a convex steepest ascent algo- rade-off between the d the accuracy of the ists of two nested it- sible to gradually de- is sufﬁciently large, aximization of F (s) ill-conditioned, then application of A† ampliﬁes the error and results in poor reconstruction, even using the Robust SL0 proposed in [9]. Introducing a regularization term in the optimization problem enables a stable recovery of x = Ds. As in the SL0 algorithm, we approximate the l0-norm by using (2), and the algorithm again consists in two nested it- erations. The internal loop seeks the maximum of F in the feasible set {s| k y As k2 ✏}. At each step we compute ˜s = s µ k and project ˜s by solving min ˆs k ˆs ˜s k2 s.t. k Aˆs y k2 ✏. (5) Using the Lagrangian function of Eq. (5), the problem can be rewritten as min ˆs k Aˆs y k2 2 + k ˆs ˜s k2 2, (6) where is the regularization parameter. The solution is ˆs = ˜s AT (AAT + IM ) 1 (A˜s y). (7) As for the SL0 algorithm, for large values, the solution is equal to the l2 norm solution subject to k y As k2 ✏. Solving the problem 2 2
- 22. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Reconstructionalgorithm Roberto Rinaldo August 31st, 2016 equals the min- x [3]. Therefore, ss is usually set n the feasible set m, and updating s µ , where: (5) he convex set to : ). (6) than BP, while However, in the thm needs to be will propose a orithms. s0 = AT (AAT + Im) 1 y; (12) The proposed algorithm is summarized in 1. Algorithm 1 -SL0 Input: µ step size, y, A, dec, min, , Kiter Initialization: s0 AT ((AAT ) + I) 1 y, 1 = 2| max(s0)| while k < min do for k=1:Kiter do s[e s2 1 2 2 k , . . . , e s2 K 2 2 k ]T s s µ Project s onto the feasible set: {s| k As y k2 ✏} s s AT ((AAT ) + I) 1 (As y) end for k k dec ˜sk s end while Output: sOUT ˜sk
- 23. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Outline • Overview of Sparse Representations and Compressive Sensing (CS) • Gaussian Dictionary for ECG approximation and CS applied to ECG signal • Analysis of non invasive Fetal Electrocardiogram (fECG) - adopted methodologies • Reconstruction algorithm • Results Roberto Rinaldo August 31st, 2016
- 24. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Results:simulatedfECG Roberto Rinaldo August 31st, 2016 Fig. 1. Reconstruction SNR versus input SNR obtained from 100 trials for simulated fECG signals, at CR=50%, using the SL0, SL0 and BPBN (SPGL1) algorithms using the Wavelet and the Gaussian Dictionary. CR 0.3 0.4 0.5 0.6 0.7 0.8 AverageReconstructionSNR[dB] 0 10 20 30 40 50 λSL0 - Gaussian Dic. SL0 - Gaussian Dic. BPDN - Gaussian Dic. λSL0 - Wavelet Basis SL0 - Wavelet Basis BPDN - Wavelet Basis Fig. 2. Reconstruction SNR versus CR obtained from 100 tri- als for simulated fECG signals. 4.1. fECG Reconstruction and Fetal Beats Detection In [7] a framework for the compression of multichannel ab- dominal fECG and joint detection of fetal beats has been pro- posed. The compression of the signal is based on Compres- sive Sensing and uses a binary sparse sensing matrix, con- taining only d = 2 ones in random positions in each column, in order to reduce the sensor complexity [8]. Before recon- struction using SL0, Independent Component Analysis (ICA) 3. PERFORMANCE OF SL0 section, the effect of noise on the reconstruction per- nce is experimentally analyzed. We compare the per- nce of the proposed algorithm with the original SL0 e BPDN-SPGL1 algorithms. The signals used in these ments are simulated fECG signals [10] with length 256. As sparsifying dictionaries we use a dictionary ussian like functions [7], and the Wavelet basis with chies’ length-4 ﬁlters. The sensing matrix elements awn as independent Gaussian random variables [2]. We the experiment 100 times with different source signals erent noise levels, and using each time a different ran- SNRin [dB] 10 20 30 40 50 AverageReconstructionSNR[dB] -10 0 10 20 30 40 50 SL0 Gaussian Dic. λSL0 Gaussian Dic. SL0 Wavelet Basis λSL0 Wavelet Basis BPDN Wavelet Basis BPDN Gaussian Dic. Fig. 1. Reconstruction SNR versus input SNR obtained from Reconstruction SNR versus input SNR obtained from 100 trials for simulated fECG signals*, at CR=50%, using the SL0, λSL0 and BPDN algorithms using the Wavelet and the Gaussian Dictionary, N=256. Reconstruction SNR versus CR obtained from 100 trials for simulated fECG signals, CR=50%. *Behar et al., “An ECG simulator for generating maternal- foetal activity mixtures on abdominal ecg recordings,” Physiological measurement, vol. 35, no. 8, p. 1537, 2014.
- 25. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Results:PhysionetChallengedatasetA* Roberto Rinaldo August 31st, 2016 Fig. 3. Sparse decomposition of the independent component in (a) using the SL0 algorithm, for (b) CR=75% and (c) CR=40% and (d) using the SL0 algorithm for CR=40%. In the graphs, different intensities represent the weight of the ac- tivated atoms. Compression Ratio [%] 20 40 60 80 100 AveragesensitivityS[%] 0 20 40 60 80 100 SL0 λ-SL0 (a) Compression Ratio [%] 20 40 60 80 100 AveragePRD[%] 0 10 20 30 SL0 λ-SL0 (b) Fig. 4. (a) Detection performance for SL0 and SL0 algo- rithm. The vertical coordinate gives the average Sensitivity for dataset A at different CR values. (b) Comparison of aver- age PRD when using the two algorithms at different CRs. As an example, we show in Fig. 3 (a) a portion of the IC of the expe matrix ( badly an value is that the almost i at lower In Fig. 4 outperfo struction 4.2. Inﬂ In this s tion pro ent sens variable not theo non-zero imentall (a) Detection performance for SL0 and λSL0 algorithm.The vertical coordinate gives the average Sensitivity for dataset A at different CR values. (b) Comparison of average PRD when using the two algorithms at different CRs. e, a second pending on recordings. ks of length , with only position is particularly an efﬁcient we consider M = 62 o CR=75%. 7 additions. sed on the hat it is the quality of , in Section framework ed with 16 signal ICs ese can be xing matrix work we use e Smoothed performance ﬁgures usually applied for the assessment of QRS detection algorithms, i.e., sensitivity (S) and positive predic- tivity (P+). According to the American National Standard [36] S and P+ are computed as S = TP TP + FN 100, P+ = TP TP + FP 100, (15) where TP is the number of true positives, FP of false positives and FN of false negatives. A detected beat is considered to be true positive if its time location differs less than 50 ms from the reference markers (within a window of 100 ms centered on the reference marker). The algorithm accuracy can be also evaluated using the F1 measure, proposed in [37], F1 = 2 S P+ S + P+ 100 = 2 TP 2TP + FN + FP 100. (16) Additionally, we apply the scoring methods proposed in [4], using two metrics, i.e., fetal heart rate measurement and RR interval measurement. The ﬁrst one, denoted here as HRmeas (bpm2 ), is used to assess the ability of the algorithm to provide valid fHR estimation. It is based on the squared difference between matched reference (fHR) and detected fHRd measurements every 5 s (12 instances for 1 min long signals) HRmeas = 1 12 12X (fHRi fHRd i )2 . (17) We repeat the experiment 20 times with different random sparse binary matrix (d = 2), for all the signals in dataset A. The reported values are the average of these simulations. *“Physionet challenge 2013,” http: www.physionet.org/challenge/2013/.
- 26. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Results:sensingmatrix Roberto Rinaldo August 31st, 2016 Table 1. Average performance of detection and reconstruction for SL0 and SL0 for dataset A. SL0 SL0 CR S PRD S PRD % [%] [%] [%] [%] 40 Sparse 2 46 5.27 85 3.77 Gaussian 45 5.58 84 3.72 50 Sparse 2 77 5.93 85 4.49 Gaussian 75 5.71 85 4.27 75 Sparse 2 84 8.81 84 8.14 Gaussian 84 8.80 84 8.02 retical reconstruction performance for i.i.d. Gaussian sensing matrices is well established, we can see experimentally that, for the class of signals we are considering, sparse matrices have similar performance. The use of a sparse sensing matrix with d = 2 allows to achieve almost identical reconstruc- tion results, besides the very low complexity implementation. Finally, Table 1 summarizes the average reconstruction and can efﬁcie beats in th ratios. Mo ing matric compares permitting [1] D. L ory, 1306 [2] E. J ceed cian [3] D. C sign IEEE [4] G. D Average performance of detection and reconstruction for SL0 and λSL0 for dataset A.
- 27. UNIVERSITY OF UDINE – ITALY – DPIA Eusipco 2016, Budapest www.uniud.it Conclusions The proposed method has been tested on public datasets (set A and set B of the Physionet Challenge, Silesia dataset*), showing promising results for both reconstruction quality and detection/classification performance The use of the proposed λSL0 reconstruction algorithm is crucial for consistent performance at all compression ratios Experiments show an average reconstruction time for the λSL0 algorithm ranging from 0.07 s, when CR=30%, to 0.01 s, when CR=80%. Thus, it maintains approximately the same computational cost of the original SL0 algorithm (ranging from 0.03 s to 0.01 s), while being much faster than the BPDN algorithm (1.6 s to 0.6 s). Programs are written in Matlab, running on an Intel Core i7 processor, equipped with 16 GB memory. The proposed framework has good performance and is suitable for real-time implementation with low-power sensors and low complexity devices. Roberto Rinaldo August 31st, 2016 *A. L. Goldberger, L. A. Amaral, L. Glass, J. M. Hausdorff, P. C. Ivanov, R. G. Mark, J. E. Mietus, G. B. Moody, C.-K. Peng, and H. E. Stanley, “Physiobank, physiotoolkit, and physionet components of a new research resource for complex physiologic signals,” Circulation, vol. 101, no. 23, pp. e215–e220, 2000.

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