compressed sensing based method for ecg compression - MIRLab

COMPRESSED SENSING BASED METHOD FOR ECG COMPRESSION Luisa F. Polania, Rafael E. Carrillo, Manuel Blanco-Velasco† and Kenneth E. Barner 

Dept. of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 † Dept. Teor´ıa de la Se˜nal y Comunicaciones, Universidad de Alcal´a, Madrid, Spain ABSTRACT

Compressive sensing (CS) is a new approach for the acquisition and recovery of sparse signals that enables sampling rates significantly below the classical Nyquist rate. Based on the fact that electrocardiogram (ECG) signals can be approximated by a linear combination of a few coefficients taken from a Wavelet basis, we propose a compressed sensing-based approach for ECG signal compression. ECG signals generally show redundancy between adjacent heartbeats due to its quasi-periodic structure. We show that this redundancy implies a high fraction of common support between consecutive heartbeats. The contribution of this paper lies in the use of distributed compressed sensing to exploit the common support between samples of jointly sparse adjacent beats. Simulation results suggest that compressed sensing should be considered as a plausible methodology for ECG compression. Index Terms— Compressed sensing, signal reconstruction, ECG compression, wavelet transform. 1. INTRODUCTION The electrocardiogram is widely used because it is a noninvasive way to establish clinical diagnosis of heart diseases. Long-term records have become commonly used to detect information from the heart signals. In these cases, the quantity of data grows significantly and compression is required for reducing the storage and transmission times. ECG compression techniques have been classified into three categories [1]: Direct methods, transform methods and other compression methods. In the second category, the wavelet transform-based algorithms have received a great deal of attention because of their straightforward implementation, and their good localization properties in time and frequency domains. The latter works in this area are characterized by hierarchical tree structures, such as embedded zero-tree wavelet (EZW) [2] and set partitioning in hierarchical tree (SPIHT) [3] protocols, which make use of the self-similarity of the wavelet transform across scales within a hierarchically decomposed wavelet tree. In recent years, CS theory [4] has generated significant interest in the signal processing community because of its potential to enable signal reconstruction from significantly fewer data samples than suggested by conventional sampling theory.

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This revelation is promising for applications in signal acquisition and compression. Compared to conventional ECG compression algorithms, CS has some important advantages: 1) It transfers the computational burden from the encoder to the decoder, and thus offers simpler hardware implementations for the encoder; 2) the location of the largest coefficients in the wavelet domain does not need to be encoded. Some recent biomedical applications include CS for rapid magnetic resonance imaging [5], and CS of electroencephalogram signals [6]. In a recent work [7], a modified simultaneous matching pursuit for Multi-lead ECG data compression is proposed, in which the idea is to exploit the correlation between various signal channels. In this work, we propose a single-lead compression method. The idea is to exploit the quasi-periodic nature of the ECG signal, shown in the correlation between samples of adjacent beats. We show that the fraction of common support is high between adjacent beats, which demonstrates that they are jointly sparse and suitable for distributed compressed sensing [8]. Our method starts with a preprocessing stage that detects the peaks of QRS complexes and makes the period of each beat constant. Then, the simultaneous orthogonal matching pursuit algorithm [9] is modified to incorporate the ideas of partially known support presented in [10]. The performance of the proposed algorithm in terms of reconstructed signal quality is evaluated using the MIT-BIH Arrhythmia Database. 2. BACKGROUND AND MOTIVATION 2.1. Compressed Sensing Review Let x ∈ RN be a signal that is either K-sparse or compressible in some orthogonal basis Ψ, then x can be well approximated by a linear K combination of a small set of vectors from Ψ, i.e. x ≈ i=1 si ψi , where K  N . Let Φ be an m × N sensing matrix, m < N . Compressive sensing [4] deals with the recovery of x from undersampled linear measurements of the form y = Φx = ΦΨs. Compressed sensing states that when the columns of the sparsity basis Ψ cannot sparsely represent the rows of the measurement matrix Φ and the number of measurements m is greater than O(Klog(N/K)), then it is possible to recover the original signal. In practical scenar-

ICASSP 2011

Fig. 1. Block Diagram of the proposed method ios with noise, the signal s can be recovered by solving the following convex optimization (1)

with  a bound on the measurement noise. 2.2. Joint-sparse recovery of signals The joint recovery idea is based on the concept of joint sparsity of a signal ensemble. In our setting, we use the joint sparsity model JSM-2 [8], in which all signals are constructed from the same sparse set of basis vectors, but with different coefficients. This approach enable us to exploit both intraand inter-signal correlation structures. First, each signal is independently encoded projecting it onto another, incoherent basis and then the resulting coefficients are transmitted to a single collection point. A decoder at the collection point can jointly reconstruct all of the signals. The sparse approximation can be recovered via greedy algorithms such as Simultaneous Orthogonal Matching Pursuit (SOMP) [9]. We use the SOMP algorithm in our compression framework (section 3.3). SOMP is a variant of OMP that seeks to identify the support one element at a time for all the signal ensemble. 3. METHODS The block diagram of the proposed compression scheme is presented in Fig. 1. The details of the implementation are illustrated in subsequent steps as follows. 3.1. Preprocessing We propose the incorporation of a sliding window whose fixed length is going to be determined by the joint sparsity pattern of the heart beating. In order to generate an approximate real time transmission, the length of the window should be short. At the same time, we want to incorporate many heartbeats in the window to recover the signal with fewer samples. Fig. 2 illustrates how the fraction of common support between adjacent beats varies as the number of beats increases, where the common support is defined as the intersection of the support sets of adjacent beats. The experiment

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Fig. 2. Fraction of common support as a function of the number of consecutive heartbeats was carried out over a single-lead ECG extracted from records 100, 102, 107, 109, 115 and 118 from the MIT-BIH Arrhythmia Database. The sparse representation is constructed using Daubechies wavelets (db4). Based on Fig. 2, and taking into consideration the trade-off between speed in transmission and number of samples, we decide to fix the window length to six heartbeats so that the fraction of joint sparsity exceeds 0.75. To properly exploit the interbeat dependencies, we incorporate a period normalization step. First, we identify and separate each period of the heart beating in the window. Then, we round the mean of the ECG beats in the sliding window to the nearest multiple of 2L , where L is the wavelet decomposition level. Let x = [x(0) x(1) . . . x(N0 − 1)] denote an ECG cycle. The normalized cycle, denoted as xn = [xn (0) xn (1) . . . xn (Nn − 1)], can be computed as follows: xn (m) = xˆ(t∗ ),

(2)

where x ˆ(t∗ ) is an interpolated version of the samples by using cubic-spline interpolation, and t∗ = (m ∗ N0 )/(Nn ), where N0 refers to the period of the original signal and Nn is the normalized period length. Once the 6 heartbeats of the window have been normalized, they are organized in a Nn × 6 matrix denoted by X. The original periods are sent to the decoder as side information since they are needed for the recovery stage.

3.2. Sampling and Quantization Compressive sensing directly acquires a compressed signal representation without going through the intermediate stage of acquiring N samples, where N is the signal length. The information we gather about X can therefore be described by Y = ΦX, where Φ is a m × N matrix, that needs to satisfy the restricted isometry property [4] in order to recover the best k-term approximation of the original signal. It is known that random matrices satisfy this condition with overwhelming probability. Here we assume that the entries of the matrix Φ are independently sampled from the normal distribution with mean zero and variance 1/m. After the sampling, we encode each column of Y through scalar quantization.

CR =

11 × N , (Bm × m) + (Bp × P )

(3)

where N is the length of the input signal, 11 is the number of bits to encode each sample, Bm are the bits for the measurements, m is the number of measurements, P is the number of ECG cycles and Bp are the bits for storage of each cycle. Let x and x ˆ be the N-dimensional vectors representing the original and reconstructed signals, respectively. The PRD is defined as P RD = (x − x ˆ/x) · 100, where the operator  ·  denotes the Euclidean norm. 4. EXPERIMENTAL RESULTS

3.3. Signal Recovery In this step we will modify Simultaneous Orthogonal Matching Pursuit by incorporating information related to the structure of the wavelet coefficients. The low-pass approximation of the first subband accumulates the majority of the energy of the signal. Therefore, it is natural to use the positions of these coefficients to select the columns of the matrix and subtract off its contribution to the measurement matrix before starting to iterate. The entire algorithm is specified in Algorithm 1. The output corresponds to the matrix S whose columns are the sparse representation of each normalized cycle in the window, and therefore X = ΨS. The original periods can be recovered by using the same transformation in (2). Algorithm 1 SOMP Algorithm with partially known support Require: Matrix Θ = ΦΨ, Matrix of measurements Y and partial known support T0 . 1: Initialize i = 0 2: Θ0 is constructed by selecting the columns of Θ in T0 . 3: Solve least squares problem S0 =arg minS Y −Θ0 S2F , where  · F denotes the Frobenius norm of a matrix 4: R0 = Y − Θ0 S0 . 5: while halting criterion do 6: i = i + 1. J 7: Find λi =arg maxj k=1 |Θj , Rt−1 ek |, where ek denotes the k-th canonical basis vector in RJ 8: Augment the index set Λi = Λi−1 ∪ {λi } and the matrix of chosen atoms Θi = [Θi−1 Θλi ] 9: Solve Si =arg minS Y − Θi S2F 10: Ri = Y − Θi Si 11: end while ˆi 12: return S ← S

3.4. Performance measure The compression ratio (CR) and the percentage root-meansquare difference (PRD) will be used as a performance measure. The CR is computed as follows:

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Experiments are carried out over a 10-min long single-ECG lead from the MIT-BIH Arrhythmia Database. For our simulations, we average 100 repetitions of each experiment. Fig. 3 shows the performance of the proposed compressor compared with SPIHT for ECG records 100, 107, 115 and 117. The record 115 is included in the set to evaluate the performance of the algorithm in the case of irregular heartbeats. We use the Daubechies db4 wavelets, four bits for the quantization of the measurements, eight bits for the beat periods and decomposition level L = 4 since it gave the best sparse representation of the ECG signals in our experiments. The proposed method achieves a good compression ratio for low PRDs that correspond to high quality in the reconstruction and even outperforms SPIHT as the PRD increases. Figure 3 also illustrates the inability of our approach to achieve a very low PRD, in the range 0-2, when the number of measurements increases due to the fact that only four bits are employed. The waveforms of the record 117 given by the proposed compression scheme are visually evaluated in Fig. 4. The reconstructed signal remains close to the original signal and the error, which is depicted for a longer range of samples, is equally distributed along the horizontal axis. This implies that the proposed method performs well locally and does not incorporate outliers in the reconstruction. 5. CONCLUSIONS In this paper, a new approach for the storage of ECG signals based on compressed sensing is presented. Simulations show that random measurements provide a plausible method of representing ECG signals. Unlike state-of-the-art algorithms for ECG compression, our method does not need to encode the location of the largest coefficients in the wavelet domain and offers a low-complexity encoder. Future work in this area is promising since the incorporation of entropy coding, optimal quantization and adaptive sampling methods based on the structure of the sparsity pattern of the ECG signal can improve the compression ratio significantly.

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[4] E.J. Cand`es and M.B. Wakin, “An introduction to compressive sampling,” Signal Processing Magazine, IEEE, vol. 25, no. 2, pp. 21 –30, Mar. 2008.

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[5] M. Lustig, D. Donoho, and J.M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” MRM, vol. 58, no. 6, pp. 1182–1195, Dec. 2007.

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Fig. 3. Comparison of our proposed method using Compressed Sensing with SPIHT. Compression ratio as a function of the PRD for different ECG records. (a) 117, (b) 115, (c)107 and (d)100.

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[6] S. Aviyente, “Compressed sensing framework for eeg compression,” in Proc. of 14th IEEE Workshop on SSP, Madison, WI, Aug. 2007, pp. 181 –184.

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Fig. 4. Compression waveform of record 117 for CR=7.23 PRD=2.57. (a) Original signal. (b) Reconstructed signal. (c) Error signal. 6. REFERENCES ´ Bravo, [1] M. Blanco-Velasco, F. Cruz-Rold´an, F. L´opez, A. and D. Mart´ınez, “A low computational complexity algorithm for ECG signal compression,” Medical Eng. Phys., vol. 26, no. 7, pp. 553 –568, Sept. 2004. [2] M.L. Hilton, “Wavelet and wavelet packet compression of electrocardiograms,” IEEE T-BME, vol. 44, no. 5, pp. 394 –402, May 1997. [3] Z. Lu, D. Y. Kim, and W. A. Pearlman, “Wavelet compression of ecg signals by the set partitioning in hierarchical trees algorithm,” IEEE T-BME, vol. 47, no. 7, pp. 849 –856, July 2000.

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[7] Q. Tan, B. Fang, and P. Wang, “Improved simultaneous matching pursuit for multi-lead ecg data compression,” in ICMTMA, Changsha, China, Mar. 2010, vol. 2, pp. 438 –441. [8] M.F. Duarte, S. Sarvotham, D. Baron, M.B. Wakin, and R.G. Baraniuk, “Distributed compressed sensing of jointly sparse signals,” in Proc. of the 39th ACSSC, Pacific Grove, CA, Oct. 2005, pp. 1537 – 1541. [9] J.A. Tropp, A.C. Gilbert, and M.J. Strauss, “Simultaneous sparse approximation via greedy pursuit,” in Proc., IEEE ICASSP, Philadelphia, PA, Mar. 2005, vol. 5, pp. v/721 – v/724 Vol. 5. [10] R. E. Carrillo, L. F. Polania, and K. E. Barner, “Iterative algorithms for compressed sensing with partially known support,” in Proc., IEEE ICASSP, Dallas, TX, Mar. 2010, pp. 3654 –3657.