Keywords

1 Introduction

According to the World Health Organizations (WHO), cardiovascular diseases (CVD) is the number one cause of early deaths in human globallyFootnote 1. Recently, new technologies such as low cost wireless body area networks (WBAN) [2] enable the real-time ambulatory electrocardiogram (ECG) monitoring for healthcare service providers and users. However, a major challenge in such a system is due to the sensor nodes which lack energy efficiency for continuous WBAN-based ECG monitoring systems as huge amount of data are collected by them.

Compressed Sensing (CS) is a simultaneous signal acquisition and reconstruction technique from a relatively smaller set of linear projections in the transform domain [1]. CS has been applied in real-time ECG data compression for WBAN applications [6, 9, 15, 16] which offers significant advantages by low complexity data encoding and thereby making the WBAN system highly energy efficient.

Multichannel ECG (MECG) data provide more feature information, which can be very useful for diagnostic purpose compared to single-channel ECG [13]. MECG signals in different channels have both spatial and temporal correlations either in the wavelet domain or in the time domain as reported in [12]. Thus, exploiting spatial and temporal correlations of MECG signals simultaneously gives better reconstruction of signals.

CS-based ECG compression works reported in the literature have exploited either temporal [6, 8, 9, 15, 16] also called as single measurement vector (SMV) approach or spatial [5, 11] correlations also called as multiple measurement vector (MMV) approach but very few works have been reported that exploit both types of correlations [12].

In this paper, we have employed a CS model which takes advantages of spatial as well as temporal correlations simultaneously while modeling a block-sparsity property of MECG signals. A novel block sparsity-based \(l_{2}/l_{1-2}\) minimization model is proposed for MECG reconstruction. Main contributions of this work are as follows:

Contributions

  • The proposed algorithm exploits both temporal as well as spatial correlations of MECG signals in the wavelet domain.

  • A rakeness-based sensing matrix suitable for MECG acquisition and a mix norm called \(l_{2}/l_{1-2}\) for CS reconstruction are applied for effective MECG reconstruction.

  • Block-sparsity of MECG signals is exploited for CS reconstruction.

  • Simulation results are evaluated based on Physikalisch-Technische Bundesanstalt (PTB) diagnostic ECG database. Performances of the proposed method are compared with traditional CS techniques.

The rest of the paper in order as follows: Sect. 2 gives a short background on CS based MECG compression. The proposed work is explained in Sect. 3 followed by experimental results in Sect. 4. Finally, conclusions are drawn in Sect. 5.

2 Background

MECG signals are not strictly sparse, but can be sparsified in the wavelet domain because most of the wavelet coefficients have very low magnitudes, near to zero for the finest scale. The compressed measurements of sparse MECG signals are obtained as

$$\begin{aligned} \mathbf Y =\varvec{\varPhi }\mathbf {X}= \varvec{\varTheta }\varvec{\alpha } \end{aligned}$$
(1)

where \(\varvec{\varTheta }=\varvec{\varPhi }\varvec{\varPsi }\); \(\varvec{\varPsi }\, \epsilon \, \mathbb {R}^{N \times N}\) is an orthonormal matrix, \(\varvec{\varPhi }\, \epsilon \, \mathbb {R}^{M \times N}\) the sensing matrix, \(\mathbf X \, \epsilon \, \mathbb {R}^{N \times L}\) the MECG signals from L channels, \(\mathbf Y \, \epsilon \, \mathbb {R}^{M \times L}\) the compressed measurement vector, and \(\varvec{\alpha }\, \epsilon \, \mathbb {R}^{N \times L}\) the unknown sparse coefficient vector, respectively.

In [7], authors shows that rakeness-based CS (Rake-CS) is suitable for ECG signals and similar to binary sensing matrices. The prime advantage of the Rake-CS compared to the general CS techniques is that the amount of projections needed for a certain quality of service without changing the adopted family of architectures for the encoder stage, and compatibility with the hardware-friendly constraint of having \(\varvec{\varPhi }\) made only of antipodal symbols. Employing a low cost compression scheme will help in saving node power consumption in a WBAN system and in turn will extend the network lifetime supported by battery power.

3 Proposed Methodology

3.1 CS-based MECG Data Compression

We used rakeness-based sensing matrix for efficient encoding of MECG data. In this approach, the sensing matrix \(\varvec{\varPhi }\) is generated in such a way that randomness is imposed on each row of \(\varvec{\varPhi }\) keeping in view that total energy of the signal is concentrated within a few samples of the signal and at the same time ensures that mutual coherence between the representation basis \(\varvec{\varPsi }\) and \(\varvec{\varPhi }\) improves.

Fig. 1.
figure 1

Antipodal sensing matrix based on: (a) random and (b) rakeness projections from MECG signals

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Figure 1(a) shows a completely random sensing matrix and Fig. 1(b) shows a rakeness-based sensing matrix simulated for MECG signals considered for our simulations. Each matrix is of size \(224 \times 512\). It is clearly observed that the latter follows some periodic structures as opposed to the former, which further reduces the number of projections of the input for CS reconstruction. This correlation structure is also same as that of the MECG signals.

3.2 MECG Reconstruction Using Block-Sparsity Based \(l_{2}/l_{1-2}\) Minimization

A WBAN-based ECG healthcare monitoring system having multiple sensors continuously acquires and transmits signals in real time. Our aim is efficient compression and reconstruction of MECG signals with less level of distortions. Since there exists spatio-temporal correlations of MECG signals in the wavelet domain, we need to impose block sparsity property in \(\varvec{\alpha }\). Mathematically, we solve the following block \(l_{2}/l_{1-2}\) minimization problem:

$$\begin{aligned} min{ \frac{1}{2}\parallel \mathbf Y -\varvec{\varTheta }\varvec{\alpha }_{}\parallel _2 ^2 +\,\lambda (\parallel \varvec{\alpha }_{g_{i}} \parallel _{2,1}-\parallel \varvec{\alpha }_{g_{i}} \parallel _{2})} \end{aligned}$$
(2)

where the second term indicates the \(l_2/l_{1-2}\) -norm, which heavily relies on the sensing matrix with different block coherence to function well, and therefore reconstruction of signals will result in less distortions [14], \(\lambda \) is a positive regularization parameter to achieve a tradeoff between sparsity and data consistency and \(g_i \in \{ g_{1},\,g_{2},\,.......,g_{p}\}\) is the index corresponding to the \(i^\text {th}\) group that \(\varvec{\alpha }_{g_i }\in \mathbb {R}{^{{(N \times L)}_{i}} }\) denotes the subvector of \(\varvec{\alpha }\) indexed by \(g_i\). Let us assume there are p blocks with block size \(b = N/p\) in \(\varvec{\alpha }\). If \(\varvec{\alpha }\) has at most s nonzero blocks, we refer to it as the block s-sparse signal. An accurate sparse representation improves the signal reconstruction further at a given number of compressed measurements. In order to solve the model given in Eq. 2, we use the alternating direction method of multipliers (ADMM) algorithm for block sparsity detailed in [14]. The estimated coefficient matrix \(\varvec{\alpha }\) consists of wavelet domain MECG signals which can be transformed back into time domain using inverse wavelet transform.

4 Experimental Setup and Results

All experiments are performed in the MATLAB (R2015a) computing environment on a computer with 3.40 GHz-i7 core CPU, 10 GB of RAM. Dataset used for experiments are obtained from PTB diagnostic MECG database [4]. It consists of 15-channel MECG datasets from 290 patients and each signal is sampled at fs = 1 kHz with 16-bit resolution. To evaluate the performance of the proposed method, a dataset that comprises of 10 ECG records: s0014lrem, s0015lrem, s0016lrem, s0017lrem, s0020arem, s0021arem, s0027lrem, s0028lrem, s0029lrem, and s0031lrem is formed. ECG matrix \(\mathbf X \in \mathbb {R}{^{512 \times 10} }\) has been built from ten channels (leads I - V4) present in each record for our experiments. Performance of the proposed reconstruction algorithm are evaluated based on percentage root-mean-square difference (PRD) and signal-to-noise ratio (SNR) [5]. For ECG reconstruction quality, in [17] authors had classified quality of ECG signals based on the values of PRD and SNR.

To generate the sensing matrix by rakeness method, we use the gaussian random vector thresholding technique [7, Sect. IIA] and apply the source codes available atFootnote 2. In each experiment, all the CS algorithms use the same sensing matrix to compress MECG recordings. For SOMP implementation, we use the solver “somp” from the simultaneous sparse approximation toolbox [10] and for BSBL we use codes available atFootnote 3. Daubechies-6 (“db6”) wavelet [3] is used as orthonormal bases \(\varvec{\varPsi }\) for representation of ECG signals which has six vanishing moments.

Fig. 2.
figure 2

Average PRD of proposed method versus block size for MECG signals

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Table 1. Channelwise average PRD of different methods at different measurements
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Figure 2 describes the average PRD against varying block sizes, i.e. 4, 8, 16, 32, 64 (MECG signals segment length is 512) and we can see that a block size of 8 gives the best results over different block sizes in terms of PRD value. Averaged PRD values per channel using the proposed algorithm, the BSBL and the SOMP at different number of measurements for 10 channels of PTB database are shown in Table 1. The proposed algorithm performs better than the BSBL and the SOMP with the improvement of average PRD at almost all levels of measurements for different channels. This is because the proposed algorithm improves the reconstruction quality by exploiting inter- and intra-channel correlations simultaneously.

Fig. 3.
figure 3

Average PRD and SNR at different number of measurements for different algorithms

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Fig. 4.
figure 4

Reconstructed signals using the proposed method for different channels at M = 1536. Original MECG signals (4096 samples) are from PTB dataset - s0014lre

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Averaged PRD and SNR over 10 channels are also plotted in Fig. 3 for different number of measurements. PRD values of the proposed method is less than the BSBL and the SOMP algorithms. Thus, the proposed method can significantly improve the compression ratio without sacrificing the quality of recovered signals.

Figure 4 shows that the reconstruction quality of the proposed method is better. We also show corresponding error signal plots in the figure. It is evident from different waveforms that clinical information of myocardial infarction present in the QT wave segment are well preserved in case of the proposed method. Above experimental results for PTB databases demonstrate that exploiting both spatial and temporal correlations is able to produce accurate MECG reconstruction for different levels of measurements.

5 Conclusions

In this paper, we present a novel block-sparsity based model for CS reconstruction of MECG signals. Exploiting both spatial and temporal correlations simultaneously in the wavelet domain carries a significant role in the CS reconstruction. The reconstruction quality of the propose method is found to be the best in case of minimum number of measurement.